Democrats say the tax returns provide a nexus between concerns about Mr. Trump’s personal financial conflicts of interest and his campaign’s ties to the Russian government, issues that have troubled members of the president’s own party.

“I’m for transparency,” Senator Bill Cassidy, Republican of Louisiana, told a few hundred people in a high school cafeteria in Denham Springs, La., on a recent afternoon, stopping well short of a call for legislation.

Senator Ron Wyden of Oregon, the top Democrat on the Senate Finance Committee, has led the chamber’s efforts to disclose Mr. Trump’s returns, a push he said was driven by constituent outrage. “People just kind of gasp,” Mr. Wyden said. “I can tell you it comes up at every one of my town meetings. People think it is very much linked to the Russia issue.”

In the House, Democrats have pressed the issue on several, so far fruitless fronts. Representative Bill Pascrell Jr. of New Jersey, who wrote the letter that has drawn two Republican signatures, introduced a measure that would compel the House Committee on Ways and Means to request Mr. Trump’s tax documents, as it is empowered to do.

Mr. Pascrell used a privileged resolution, a procedure that allows even a single member to bypass leadership to bring up a measure for a vote — and that was more theatrically employed by some House conservatives in the fall in an effort to oust the head of the Internal Revenue Service over objections by their leaders. Mr. Pascrell’s measure failed in a floor vote last week along party lines.

Representative Kevin Brady, Republican of Texas and the chairman of the House Ways and Means Committee, has resisted calls to, as he described it to reporters, “rummage around in the tax returns of the president.”

A few Republicans have broken from their party on investigating connections between Mr. Trump’s campaign and Russia. Representative Darrell Issa of California, who narrowly won re-election in November and is considered vulnerable in 2018, recently called for an outside investigation. The House and Senate intelligence committees will conduct their own inquiries.