Mr. Santos, who was elected in November to represent a district that includes parts of Long Island and Queens, has insisted that he would not step aside despite the mounting pressure coming from several investigations and numerous calls from his House colleagues for him to resign.

The timeline for any case against Mr. Santos is still unknown. But even if Mr. Santos was convicted of a crime, he could continue to serve in Congress. He would only be removed from office if two-thirds of the members of the House of Representatives voted to expel him.

Speaker Kevin McCarthy of California, who is overseeing an exceedingly tight Republican majority, said he would call for Mr. Santos to resign if he was found guilty, but not before.

Mr. McCarthy noted that he had already asked the congressman not to serve on committees, but said that Mr. Santos deserved a right to fight the charges like other lawmakers who had been charged in the past.

“We will just follow the same pattern we always have,” he said. “If a person is indicted, they are not on committees, they have the right to vote but they have to go to trial.”

“In America, you are innocent until proven guilty,” he added.

Some of Mr. McCarthy’s Republican colleagues, however, were less generous in their views of Mr. Santos. Already, about a dozen Republicans have joined numerous Democrats in demanding that Mr. Santos resign. Many of them doubled down on their calls for action on Tuesday.

“There’s a clock ticking, and George Santos should have resigned in December,” Representative Marc Molinaro, a Republican of New York, said. “He should have resigned in January. He should have resigned yesterday, and maybe he’ll resign today. But sooner or later, honesty and justice will be delivered.”