“I cannot say that he can count on the support of Democrats,” Rep. Chris Van Hollen (D., Md.) told reporters on Friday, adding he hadn’t decided how he would vote in such a situation. “The Republican caucus should find its leader, and we stand ready” to work with whoever that is, he added.

On the other hand, Democrats know that voting with the conservatives to topple Mr. Boehner could mean paving the way for a more conservative replacement.

“When people say that Boehner should be replaced, my question is always: with whom? Someone to the far right?” asked Rep. Jim McGovern (D., Mass.), who noted that Mr. Boehner has often turned to Democrats to keep the government functioning. “There’s a better chance of working with him than somebody on his far right who doesn’t believe government should exist.”

One option is that Democrats could simply vote “present” or not vote at all, and thus avoid explicitly backing Mr. Boehner. That would leave his fate to be determined by Republican votes alone.