“With all due respect, I completely disagree with Representative Fredette’s position,” the Republican leader, Michael Thibodeau, said in a statement. “The Republican senate caucus has clearly stated that we need an acceptable plan for corrective action before the determination of whether the Legislature should convene is made.”

Mr. Thibodeau said many Senate Republicans were still deciding whether or not Mr. LePage’s apology and promise to seek “spiritual guidance” had allayed their concerns about the governor — a process that could deepen their divide with Republicans in the House and pull the controversy into a second week.

“There should be consequences for our actions,” Mr. Thibodeau said, “and I think that’s why we’re really struggling.”

There is no precedent for such a situation here — a fact that has sent lawmakers like Senator Roger Katz, a Republican who has been critical of Mr. LePage, back to reading their state Constitutions on Wednesday to figure out what, if anything, they could do now to convene a special session. Mr. Katz, a lawyer, suggested it may require only a simple majority of the House and senate, which could reduce the House Republicans’ ability to block such a move, although he added there could be other hurdles to an agreement.

“It’s never been done before, so we’re in a little bit of uncharted waters,” said Mr. Katz, who said he wanted Mr. LePage to get “a competent evaluation and follow the recommendations of the evaluator.”

Democrats have continued to call for Mr. LePage’s ouster, and on Wednesday said they were scheduling meetings with Republican leadership to discuss how to proceed.

“Coming back in for a special session will be discussed and all options around the governor’s political future are on the table,” said the Democratic leader of the Senate, Justin Alfond.