He argued that Republican candidates should not try to run away from Mr. Trump “because he has a base that is important.”

The suggestion, by Mr. Mulvaney, that Republicans might fare better without Mr. Trump as the dominant factor in voters’ minds is a far cry from the president’s personal approach to the campaign. He has held rallies across the country in recent days, focusing on red states where Democratic senators are seeking re-election, and warning conservative voters in intensely personal terms that a victorious Democratic Party would try to hound him from office.

And Mr. Mulvaney’s comments came at the end of a week that dramatized just how difficult it might be to nudge any particular issue, aside from Mr. Trump, to the center of the campaign. The president began the week raging against a coming book by the journalist Bob Woodward that depicts Mr. Trump as a hapless leader atop an administration in chaos, and concluded with Mr. Trump urging the Justice Department to root out the identity of an administration official who wrote an anonymous Op-Ed in The New York Times reinforcing that picture.

Yet if he broke in places with the president’s political posture, Mr. Mulvaney, in his weekend comments, came closer to matching the private strategic thinking of Republican congressional leaders.

The budget chief, who has been seen at points as a potential White House chief of staff, acknowledged that Republicans had nominated poor candidates in some important races and might struggle to defend a huge number of open seats in the House, where dozens of Republican lawmakers decided not to run for re-election.

Democrats must gain 23 seats in the House to take control of the chamber. Senior Republican strategists have grown sharply concerned about a collection of open seats where they have put forward flawed nominees, including in Pennsylvania, New Jersey and North Carolina. Mirroring in some respects the Republican campaigns of 2010, Democratic candidates have been running on a message of blocking Republican health care and economic policies, and reining in an unpopular White House.

The Senate appears more secure for Republicans at this point. Even though they hold only a slim, 51-seat majority, Democrats are defending far more seats than Republicans, and many of the Democratic incumbents up for re-election are running in conservative states.