Currently, banks with $50 billion or more in assets — a threshold that includes many smaller regional lenders — are treated under the 2010 law as “systemically important.” That designation subjects them to more stringent regulations. The Senate bill would raise that threshold to $250 billion, leaving only a handful of the biggest banks facing the toughest oversight.
The Senate legislation would also exempt banks with less than $10 billion in assets from the so-called Volcker Rule, which prohibits banks from making wagers in their own behalf rather than for clients.
President Trump said in a statement that he would sign the Senate bill in its current form. The problem is that it doesn’t go far enough for Republicans in the House.
“Why would we look at this product in the Senate as the max that should be done?” asked Representative Bill Huizenga, a Republican from Michigan.
The House passed its own measure to overhaul Dodd-Frank last year, but it was declared dead on arrival in the Senate. Mr. Hensarling, who is retiring next year, said components of that bill must be incorporated in a final package. He cited measures that would make it easier for start-ups to give presentations about their businesses without violating Securities and Exchange Commission rules and that would expand the types of investors who could participate in private corporate fund-raising.
Mr. Hensarling said he expected to gather Republicans and Democrats from both chambers to chart a path forward. He said Speaker Paul D. Ryan, Republican of Wisconsin, had promised not to hold a vote on the Senate bill in its current form.
While the banking world welcomed the passage of the Senate bill, they hope the House will take the legislation further in the industry’s direction.
The challenge is that if House Republicans insist on a much more aggressive regulatory rollback, it probably won’t be acceptable to moderate Democrats in the Senate, where 60 votes are needed for passage.
Analysts at the financial services firm Keefe, Bruyette & Woods gave the banking bill an 80 percent chance of being passed eventually, although House Republicans could still scuttle it if they overreach.
“It would be the greatest irony if House Republicans wind up killing the bill and doing what Senator Elizabeth Warren could not,” the analysts wrote in a note to clients.
Ms. Warren, an opponent of loosening bank rules, has publicly criticized her Democratic colleagues who supported the Senate legislation. Liberal interest groups, such as the Progressive Change Campaign Committee, are planning to unveil online advertisements in the coming days accusing senators who supported the legislation of voting with Wall Street and against working families.
Some Democratic leaders worry that such tactics might endanger the seats of senators up for re-election this year in states that Mr. Trump carried in 2016.
Republicans have been watching the infighting among Democrats with a quiet sense of glee.
And while congressional Republicans are pushing to broaden the Senate legislation, they are shelving some of their previous priorities for the sake of expediency.
Mr. Hensarling, for instance, has long wanted to dismantle the Consumer Financial Protection Bureau, the brainchild of Ms. Warren. But on Thursday he signaled that he wouldn’t insist on that being included in the current legislation.
“I vote for a lot of things I’m not comfortable with,” Mr. Hensarling said. “Right now, I’m very happy” with the bureau under Mick Mulvaney, the acting director, who has vowed to soften the agency’s approach to the banking industry.
Mr. Hensarling added: “May he rule forever.”