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.