WASHINGTON — President Trump suggested on Thursday that he might veto a bipartisan surveillance bill, potentially disrupting an agreement to resolve a debate over national security and privacy before three F.B.I. tools for investigating terrorism and espionage expire on Sunday.

“Many Republican Senators want me to Veto the FISA Bill until we find out what led to, and happened with, the illegal attempted ‘coup’ of the duly elected President of the United States, and others!” Mr. Trump wrote on Twitter.

Mr. Trump’s comments came a day after the House passed a bipartisan bill to extend the expiring tools while also adding safeguards to national-security wiretapping under the Foreign Intelligence Surveillance Act, or FISA.

Several of the president’s most vocal allies backed the legislation, and Senator Mitch McConnell, Republican of Kentucky and the majority leader, enthusiastically urged swift passage of the House’s bill.

The president did not explain whether he was suggesting that he might not sign the bill — negotiated this week by Speaker Nancy Pelosi and Representative Kevin McCarthy of California, the House Republican leader — or a short-term extension that Republican senators were contemplating.

But the intervention showed that Mr. Trump remains an unpredictable and volatile decision maker on surveillance legal policy. Republicans in the House thought they had assurances from the White House that Mr. Trump would sign their bill before they voted.

If Mr. Trump does derail the effort by congressional leaders of both parties to get some kind of bill passed by Sunday, the F.B.I. would at least temporarily lose three powers that lawmakers created after the Sept. 11 attacks. They include the authority to get a court order for business records that are relevant to a terrorism or espionage investigation.

Mr. Trump and his supporters are invested in promulgating a conspiracy theory that the F.B.I.’s counterintelligence investigation into Russia’s efforts to manipulate the 2016 presidential election was actually a politically motivated attempt to sabotage his presidency, not a legitimate attempt to understand a foreign power’s interference in American democracy.

An investigation by the Justice Department’s independent inspector general, Michael E. Horowitz, concluded that the Russia investigation — including efforts to understand the nature of numerous links between Russia and the Trump campaign — had a lawful basis, and found no evidence that its opening was politically motivated.

Mr. Horowitz did, however, uncover serious errors and omissions in one aspect of the inquiry: investigators’ applications for permission from the FISA court to wiretap Carter Page, a former Trump campaign foreign policy adviser with many links to Russian officials.

None of the expiring tools were involved in the Page applications. But the legislation to extend them has become a vehicle for Congress to respond to the inspector general’s findings. The House bill, for example, would push the FISA court to appoint an outsider to critique the government’s arguments when a wiretap application raised serious issues about First Amendment activity, which could include political campaigns.

Even before Mr. Trump’s Twitter remark, it was uncertain whether the House’s rewrite could pass the Senate in time to prevent the expiring F.B.I. tools from temporarily lapsing. Under Senate rules, senators can express their displeasure by slowing legislation moving through the chamber for days, though not stopping it.

In this case, Senator Mike Lee, Republican of Utah, was threatening to use procedural tools to prevent passage before the Senate leaves for a weeklong recess because he does not believe the House’s language sufficiently protects Americans’ civil liberties from government spying.

Senator Rand Paul, Republican of Kentucky and another critic of government surveillance who frequently speaks with the president, wrote on Twitter shortly after Mr. Trump’s post that he would “continue to stand” with him in opposition to party leaders “trying to ram through fake FISA amendments without any real changes.”

Civil libertarians on the right and left made similar objections in the House, but several of Mr. Trump’s staunchest conservative allies, including Representatives Jim Jordan of Ohio and Devin Nunes of California, who have led the attack on the F.B.I. over the Page spying case, lent their approval to the bill. Mr. Trump’s tweet puts them all in the uncomfortable position of potentially being out of step with him.

The legislation has also received public endorsement from other close allies of Mr. Trump, including Attorney General William P. Barr and Senator Lindsey Graham of South Carolina, the Republican chairman of the Judiciary Committee.

When Mr. McConnell took to the Senate floor earlier Thursday to endorse the bill, he indicated that he knew that some civil-libertarian-minded lawmakers were objecting that the bill did not go far enough and could seek to block a swift up-or-down vote on it.

Mr. McConnell’s staff was feeling out senators to determine whether, failing passage of the bill, the Senate could unanimously pass a 45-day extension of the three F.B.I. tools.

“It is not a question of if this passes, but when,” Mr. McConnell said in remarks Thursday morning on the Senate floor. “I hope that my colleagues who may not choose to vote for this legislation will not deny this body the opportunity to renew these authorities today to prevent any lapse.”

Passage of either the bill or a short-term extension is now in doubt. The House would have to approve any short-term extension as well, and it too would require Mr. Trump’s signature, raising the possibility of a lapse on Sunday. And even if the Senate did pass a 45-day extension, there is no guarantee the House would do so anytime soon. It was bogged down Thursday with legislation to address the unfolding coronavirus pandemic, and once lawmakers leave town as soon as later Thursday, the virus could potentially keep Congress from convening for weeks.

In 2018, when another part of the complex legal architecture governing national security surveillance — a law known as Section 702 of the FISA Amendments Act, which permits warrantless wiretapping of foreigners abroad — was about to expire, the White House and Mr. Trump’s Republican allies in Congress pushed to renew it with minimal new privacy protections, turning back a push by civil libertarians to impose more significant limits.

But on the cusp of the crucial vote, Mr. Trump abruptly denounced the legislation after seeing a commentator on Fox News speak critically of it and urge him to change course, telling him that his own “woes” began with surveillance.

Mr. Trump’s unexpected intervention at the time threatened to derail the legislation his own administration had pushed for. The House speaker then, Paul D. Ryan, called Mr. Trump to explain that Section 702 involved a different kind of surveillance, and eventually got him to walk back his statement. The bill passed, and Mr. Trump signed it.

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