They don’t know how, they don’t know when, they don’t even really know why, but Republicans can hardly wait to impeach that guy in the White House.

WASHINGTON — Representative Kerry Bentivolio, a freshman Republican from Michigan, has a legislative dream. It is not to balance the federal budget, or find a way to help his ailing state or even take away money from the federal health care program, a goal that has so animated many other Republicans this summer.

Rather, Mr. Bentivolio told constituents, it is to put in motion the impeachment of President Obama. “If I could write that bill and submit it, it would be a dream come true,” he said this month.

Mr. Bentivolio may be lacking in his understanding of the technical details of the impeachment process — he has retained experts and historians to help him with that, he said — but he is hardly the only one with this desire.

While many members of Congress have used their August break to engage in conversations about immigration policy, the federal budget and the impending implementation of the Affordable Care Act, some Republicans have taken the opportunity to raise the specter of — if not quite the grounds for — presidential impeachment.

At least two other House Republicans told voters this month that the impeachment process could happen. And last week, Senator Tom Coburn, an Oklahoma Republican who has called himself a friend of the president, told constituents that the nation was “perilously close” to an impeachment situation.