By Rebecca Ballhaus, Lindsay Wise, Natalie Andrews, Wall Street Journal–

Republican leaders said they don’t currently have enough votes to block witnesses in President Trump’s Senate impeachment trial, people familiar with the matter said, after his legal team concluded its efforts to counter Democrats’ charges that the president abused power and obstructed Congress.

On the third and final day of presentations by the Trump legal team, lawyers tried to cast doubts on the importance and credibility of allegations by former national security adviser John Bolton about the president’s motives for freezing aid to Ukraine. Republicans had hoped to wrap up the trial with an acquittal of the president by this week, but Democrats have said Mr. Bolton should appear under oath to offer a firsthand account of the president’s motivations for freezing aid to Ukraine—a matter at the heart of the impeachment case.

At a meeting of all Republican senators late Tuesday, Senate Majority Leader Mitch McConnell (R., Ky.) said the vote total wasn’t where it needed to be on blocking witnesses or documents, the people familiar with the matter said. He had a card with “yes,” “no,” and “maybes” marked on it, apparently a whip count, but he didn’t show it to senators.

Sens. Cory Gardner of Colorado, Martha McSally of Arizona and Thom Tillis of North Carolina, who face competitive races in the fall, addressed their colleagues in the meeting, people familiar with the matter said. Mr. Gardner said a longer trial would lead to more Democratic attacks, according to a spokesman, and Mr. Tillis called impeachment a sham. Ms. McSally’s office said she doesn’t comment on what happens in private meetings.

An administration official said the White House was optimistic it would get the necessary votes by Friday. “We are still in the game,” the official said.

During Tuesday’s proceedings, the president’s lawyers argued that House managers hadn’t established their case that Mr. Trump abused power and obstructed Congress and said the accusations fell short of the threshold needed to remove a president from office, particularly in an election year.

“The bar for impeachment cannot be set this low,” said Jay Sekulow, one of the president’s personal attorneys, referring to the Democrats’ impeachment case. Deputy White House counsel Pat Philbin said the abuse-of-power article was “infinitely malleable” and allowed for too much subjectivity. “How are we supposed to get the proof of what’s in the president’s head?” he asked.

White House counsel Pat Cipollone, in closing, reminded senators that the presidential election was nine months away and said the choice should be left to voters. “Why tear up their ballots?” he asked.

Senate Minority Leader Chuck Schumer (D., N.Y.), following the conclusion of the session, dismissed the Trump legal team’s arguments. “Their whole argument is diversion,” he said.

“If you don’t believe the newspaper report, call the witnesses,” he said of the Bolton account, which was first reported Sunday in the New York Times. He reminded senators that the witnesses Democrats have wanted to call but that have been blocked by the White House—which include Mr. Bolton and acting White House chief of staff Mick Mulvaney—are Republicans appointed by the president.

Democrats, who control 47 seats, need four Republicans to join them to approve motions for new testimony or documents, which need a simple majority to pass. Starting Wednesday, the Senate will have two days to ask each side questions, followed by a vote later this week on new evidence.

In Tuesday’s closing, the defense team was seeking to persuade senators not to support hearing testimony from further witnesses, among them Mr. Bolton. Mr. Bolton wrote in a draft of his forthcoming book that the president told him he wanted to keep aid to Ukraine frozen until Kyiv had aided investigations into Democrats, including former Vice President Joe Biden and his son, Mr. Bolton’s lawyer confirmed.

The president has denied Mr. Bolton’s description of the conversation. He has repeatedly said there was no link between freezing the aid to Ukraine and his push for investigations.

The president’s lawyers on Tuesday argued that too little is known about the Bolton claims for them to factor into the Senate trial.

Mr. Sekulow called the allegations “inadmissible” and pointed to the president’s denials, while accusing the former adviser of seeking to boost his book sales. Impeachment, he added, “is not a game of leaks and unsourced manuscripts. That is politics, unfortunately.”

As the president’s allies attacked Mr. Bolton’s credibility, former White House chief of staff John Kelly defended his former colleague. “If John Bolton says that in the book, I believe John Bolton,” he told a crowd at Ringling College in Sarasota, Fla., according to local news reports.

The House last fall had sought Mr. Bolton’s testimony but moved forward with articles of impeachment when he declined to appear without a subpoena. The House declined to subpoena him in an effort to move quickly and not get bogged down in court fights.

But the reports of Mr. Bolton’s account unsettled other Republican senators. Several who are on the fence about witnesses said Mr. Bolton’s claims strengthened the case for further testimony, while the number of senators the White House believes may vote for more testimony ticked up.

One proposal by Sen. James Lankford (R., Okla.) to make the Bolton manuscript available to the Senate in a classified setting for review drew the support of Sen. Lindsey Graham (R., S.C.), a top ally of the president’s.

Mr. Schumer rejected the idea of reading the manuscript behind closed doors. “What an absurd proposal,” he said. “It is a book. There is no need for it to be read in the SCIF unless you want to hide something,” referring to a secure facility. The book is set to be published in March.

While no administration officials during the fall’s House hearings testified that they were told directly by Mr. Trump that he was holding up the aid to pressure Kyiv, four current and former officials said they understood that to be the case.

Republicans have said this week that if the Senate votes for more testimony, they want to call witnesses including Mr. Biden and his son, Hunter.

Sens. Mitt Romney (R., Utah) and Susan Collins (R., Maine) indicated on Monday that they were likely to favor witnesses. Sens. Lisa Murkowski (R., Alaska) and Lamar Alexander (R., Tenn.) remained open to the idea.

“I think that John Bolton probably has something to offer us,” Ms. Murkowski said Tuesday.

The White House has grown concerned in recent days that two other senators might vote in favor of more testimony: Pat Toomey of Pennsylvania and Rob Portman of Ohio, people familiar with the discussions said. The defense team has been preparing for the possibility that the Senate will vote in favor of calling more witnesses and discussing how it would go to court to fight a subpoena for Mr. Bolton’s testimony. Mr. Bolton has said he would testify during a Senate trial if subpoenaed.

Mr. Toomey suggested at a closed-door Senate lunch on Monday an arrangement in which the Senate subpoena Mr. Bolton as well as a witness sought by the White House, an approach Mr. Romney said he found fair.

“I think if you hear from one side, you probably ought to have a chance to hear from witnesses from the other side,” said Mr. Romney.

Republicans senators said that during the coming days of questioning they planned to ask about the elder Mr. Biden as vice president withholding loan guarantees from the Ukrainian government when the nation was slow to fire a prosecutor general whom Western diplomats criticized for not cracking down on corruption.

They also planned to ask about Hunter Biden’s role on the board of a Ukrainian gas company while his father was vice president. Mr. Trump and his allies have argued it was corrupt for Mr. Biden to call for the ouster of the Ukrainian prosecutor because he had once investigated the gas company, Burisma Holdings, that had ties to his son. The Bidens deny wrongdoing. Hunter Biden has said that serving on the company’s board showed poor judgment given his father’s anticorruption efforts.

Republicans also plan to use their questions to attack the chairman of the House Intelligence Committee, Rep. Adam Schiff (D., Calif.), over his office’s dealings with the whistleblower whose account of the Ukraine pressure campaign spurred the impeachment inquiry.

Democratic senators said they would like to ask questions that could allow the Democratic House managers to rebut comments from the Trump defense team.

Following the period of questioning, the Senate will hold as much as four hours of debate on whether to subpoena witnesses or documents. If that hurdle is cleared, the Senate would then proceed to vote on whether to hear from Mr. Bolton, subpoena the notes he took during his tenure as national security adviser and gather other information or hear from other witnesses.

Write to Rebecca Ballhaus at Rebecca.Ballhaus@wsj.com, Lindsay Wise at lindsay.wise@wsj.com and Natalie Andrews at Natalie.Andrews@wsj.com