WASHINGTON — White House officials once debated a scorched-earth strategy of publicly criticizing and undercutting Robert S. Mueller III, the special counsel investigating Russian efforts to disrupt last year’s election.

Now, President Trump’s lawyers are pursuing a different course: cooperating with the special counsel in the hope that Mueller will declare in the coming months that Trump is not a target of the Russia inquiry.

Trump has long sought such a public declaration. He fired his FBI director, James B. Comey, in May after Comey refused to say openly that Trump was not under investigation.

The president’s legal team is working swiftly to respond to requests from Mueller for e-mails, documents, and memos, and will make White House officials available for interviews.

Once Mueller has combed through the evidence, Trump’s lawyers plan to ask him to affirm that Trump is not under investigation, either for colluding with Russian operatives or trying to obstruct justice.

More than a half-dozen White House officials, witnesses and outside lawyers connected to the Russia inquiry have described the approach, which is as much a public relations strategy as a legal one.

The president’s legal team aims to argue that the White House has nothing to hide, hoping to shift the burden to Mueller to move quickly to wrap up an investigation that has consumed the Trump administration’s first year.

“The White House believes the special counsel shares its interest in concluding this matter with all deliberate speed for the benefit of the country,’’ said Ty Cobb, the White House lawyer handling the response to Mueller’s investigation.

He said the administration was cooperating “with hope of bringing the matter to a prompt and decisive end.’’

Any public declaration by Mueller about the president’s innocence would also be a clear sign that the special counsel’s investigation has not broadened significantly beyond last year’s presidential campaign to include a close scrutiny of any of Trump’s past business dealings with Russians.

Whether the strategy will work is another matter. The plan rests on the premise that Trump has done nothing wrong — something the president has repeatedly told his lawyers and said publicly — and some lawyers connected to the investigation say that Cobb has been too willing to take the president at his word.

If the White House moves too hastily, they argue, materials could end up in Mueller’s hands that might damage the president and other administration officials.

Donald F. McGahn II, the White House counsel, previously expressed fears that the document production could set a bad precedent for future administrations. Cobb has told aides that the White House should move deliberately and carefully, but not drag its feet.

Others doubt that Mueller will publicly clear Trump anytime soon even if the documents and interviews do not show that he committed a crime.

Mueller is building cases against two of Trump’s former advisers, Paul J. Manafort and Michael T. Flynn. Should either man cooperate with investigators, it might change Mueller’s view of how Trump fits into the Russia investigation.

Nevertheless, the president’s advisers have concluded that this strategy represents their best chance to lift the cloud hanging over the administration.

“Good for them if they can pull it off,’’ said Barbara Van Gelder, a prominent Washington white-collar lawyer who served in the Justice Department with Mueller. She said he was highly unlikely to give the White House any assurances as long as the investigations into Manafort and Flynn were open.

“Mueller’s not going to make a statement,’’ she said, “because he’s not going to want to claw it back.’’

In another potential sign of the White House’s willingness to compromise, Trump tweeted Saturday that he had spoken to Senate minority leader Chuck Schumer to gauge whether Democrats are interested in helping to pass ‘‘great’’ health legislation.

The New York senator answered that Democrats are willing to hear his ideas, but scrapping the Obama health law is a nonstarter.

Trump’s latest overture to Democrats follows GOP failures so far to fulfill their yearslong promise to repeal and replace the Affordable Care Act despite controlling the White House and Congress since January.

The president tweeted that he called Schumer on Friday to discuss the 2010 law, which Trump said ‘‘is badly broken, big premiums. Who knows!’’ Trump said he wanted ‘‘to see if the Dems want to do a great health care bill.’’

Schumer said through a spokesman Saturday that Trump ‘‘wanted to make another run at repeal and replace and I told the president that’s off the table.’’

Trump has said before that he would be open to talking with Democrats on health care, but there have been no clear signs of a compromise between Republicans who have sought to scrap former President Barack Obama’s law and Democrats who want to protect it.

Schumer said a starting point could be negotiations led by Senators Lamar Alexander, Republican of Tennessee, and Patty Murray, Democrat of Washington, who have been discussing a limited bipartisan deal to stabilize state-level markets for individual policies.

People covered under the health law represent about half of those who purchase individual policies.