On Thursday afternoon, the president posted his support on Twitter for the congressional negotiations. “I like Sen. Grassley’s drug pricing bill very much, and it’s great to see Speaker Pelosi’s bill today,” he tweeted. “Let’s get it done in a bipartisan way!”

Mr. Trump could also go his own way, following through on a promise to put out a proposed rule or executive order basing the prices for certain drugs in Medicare on those in other countries. That idea makes Republicans in Congress just as queasy as letting Medicare negotiate with drug companies, but it appeals to Mr. Trump’s “America First” sensibility and would be more limited.

“You’ve got the makings of a Trump-Pelosi alliance here on drug pricing, and that would be a powerful combination,” said Representative Peter Welch, Democrat of Vermont, who serves on the Energy and Commerce Committee. “Trump is the one who asked the practical question: Why do we get ripped off, why are we the suckers paying the highest prices in the world? Now he has an opportunity to address that.”

Ms. Pelosi would first have to bring both liberals and centrists in her caucus on board; she has already worked to win the liberals by dropping an initial proposal that would have let the Government Accountability Office, an independent investigative arm of Congress, decide a drug’s price if the government and manufacturer could not agree. Liberals did not want to farm that job out to a third party.

But some liberals are still unsatisfied with the number of drugs designated for annual price negotiations, for one thing; the bill would allow for a minimum of 25 per year and a maximum of 250.

“We think 25 a year is too low to have our constituents really appreciate” it, said Representative Mark Pocan, Democrat of Wisconsin and a chairman of the Congressional Progressive Caucus. He added that his group also thought the penalty for noncompliant drug companies should be even harsher, pointing to a bill from Representative Lloyd Doggett, Democrat of Texas, that would allow other companies to produce cheaper generic versions of any drug whose price the brand-name manufacturer refused to negotiate with the government.

A far bigger challenge would be getting Ms. Pelosi’s bill through the Senate, where it could die a fast death given the Republican aversion to several of its main provisions. In that, Mr. Trump is a wild card: His administration has been discussing drug pricing with Ms. Pelosi’s office for months, according to Mr. Deere.