Perhaps the most consequential concern relates to a House Republican proposal to get rid of a rule that lets companies write off the interest they pay on loans — a move real estate developers and Mr. Trump vehemently oppose. Doing so would raise $1 trillion in revenue and reduce the appeal of one of Mr. Trump’s favorite business tools: debt.

In the halls of Congress, Democrats are employing procedural maneuvers to drive home their point on the tax returns and possibly compel Republican lawmakers to join their effort to force Mr. Trump to release them. And Democratic aides say more tricks are coming.

More than a dozen Republicans — from recognizable names like Senator Joni Ernst of Iowa and Representative Mark Sanford of South Carolina to backbenchers like Representatives David Young of Iowa, Matt Gaetz of Florida, Walter B. Jones of North Carolina, Ted Yoho of Florida, Rodney Frelinghuysen of New Jersey and Justin Amash of Michigan — have agreed that Mr. Trump should release his returns.

That list grows almost daily. On Monday, former Representative Joe Walsh of Illinois, a conservative firebrand and Trump loyalist, said the president should release his tax returns. “I do think this issue will come back and bite him on the butt,” he said on MSNBC.