WASHINGTON — The impeachment inquiry that has imperiled President Trump and captured the attention of a divided nation is now threatening to overshadow House Democrats’ attempts to show that they can govern, posing a political dilemma that deepens as the investigation marches on.

Democrats returned to Washington this week determined to spotlight the popular domestic initiatives, like lowering the cost of prescription drugs and education, that helped to sweep them into the majority last year. But instead of focusing on those, their first private gathering on Capitol Hill was dominated by a debate over whether they should cast a formal vote to authorize the impeachment inquiry they began three weeks ago.

The meeting in the basement of the Capitol on Tuesday night made it clearer than ever that many Democrats — particularly those from conservative-leaning districts who flipped Republican seats and now face difficult re-election races — have no interest in spending time on the House floor calling attention to the impeachment push.

The idea of having a vote, which had been floated by leadership earlier in the day, was quickly shot down, according to a Democratic aide in the room, speaking on condition of anonymity without authorization to describe the private meeting. While many Democrats are open to a vote, nervous freshmen in swing districts were opposed, fearing it would be a distraction.