But Ms. Barzee Flores said Democrats should not treat impeachment as a singular issue in 2018, over matters like health care and immigration. “There are other issues that are pressing,” she said. “More pressing, even.”

For Mr. Steyer, however, impeachment is a singular cause. And he is no easy character for Democrats to ignore: From his headquarters in a San Francisco office tower, Mr. Steyer, 60, has built a sprawling political operation with more than 200 staff members around the country, mostly employed by his flagship group, NextGen America, which focuses on climate change.

Mr. Steyer has amassed a mixed win-loss record over the years, along with a mercurial reputation among Democratic strategists. But he is nearly alone among Democratic donors in his willingness to spend money on a titanic scale. In 2016, Mr. Steyer spent more than $90 million supporting Democrats, and his checkbook may be critical to their efforts to capture Congress.

Separate from the impeachment drive, Mr. Steyer, who made his fortune as a hedge fund investor, announced this month that he would spend $30 million on mobilizing young voters in 2018. He has already quietly funneled nearly $2 million into a union-backed effort to capture seven Republican-held seats in California, three strategists directly involved in the campaign said.

But on impeachment, Democrats in Congress have made no secret of their skepticism. Nancy Pelosi and Chuck Schumer, the Democratic minority leaders in the House and Senate, have called it a premature proposal. Senator Bernie Sanders of Vermont, a hero of the left, said Democrats should avoid “jumping the gun” on such a drastic action. And the Democratic Congressional Campaign Committee has urged candidates to use more measured language, endorsing investigations of Mr. Trump but avoiding an impeachment pledge.

Ms. Pelosi, who has a longstanding relationship with Mr. Steyer and hosted him in her private box at the Democrats’ 2016 convention, conveyed her concerns to him directly. She told Mr. Steyer in a phone call last fall that impeachment was a profoundly divisive issue and stressed that other matters, like the Republican-backed tax overhaul, were more urgent, according to four people briefed on the conversation.