But Democrats expect that Mr. Trump and his appointees within the Treasury would outright refuse to comply, tempting a lawsuit by the House and all but ensuring a long court fight over the legitimacy of Congress’s oversight of the chief executive.

Rudolph W. Giuliani, Mr. Trump’s lawyer, said in a brief interview on Tuesday that he would advise Mr. Trump to fight any such request and saw reason to think he could win. With control of the House, he said, Democrats would merely be conducting a “circus” and would probably have a difficult time proving they had any legitimate legislative or oversight objective.

“It is really for the purpose of political harassment,” Mr. Giuliani said, adding, “It is a heck of a good battle for a president.”

Democrats have publicly vowed to move forward if they gain control of the House after November’s elections. Representative Nancy Pelosi of California, the Democratic leader, told The San Francisco Chronicle last week that the public was “owed” transparency in light of Mr. Trump’s refusal to release his tax returns, a breach of modern presidential norms.

“That is one of the first things we’d do — that’s the easiest thing in the world,” she said.

Could it backfire politically?

But mindful as much of the party’s prospects in 2020 as of inflicting short-term pain on Mr. Trump, party leaders are divided on how quickly and aggressively to act. Some in the party fear that if they fail to build an adequate public rationale first, Mr. Trump would paint the majority as cravenly abusing its authority, undercutting Democratic efforts to project a steady-hand alternative to the Republicans.