Gov. Eric Greitens of Missouri resigned rather than face a felony computer tampering charge, depriving Democrats of a political weapon they had hoped to wield in the Senate race there. (A felony invasion of privacy charge against Mr. Greitens, who was accused of sexual misconduct, was dropped weeks earlier.) And the ailing Senator John McCain remains in office, passing a crucial deadline that all but ensures there will be only one Senate seat up for grabs in Arizona.

“The Republican caucus in the Senate is feeling substantially more optimistic now than at this time last year,” said Senator Roy Blunt of Missouri, predicting his party will gain a handful of seats.

Not everyone in the G.O.P. is as bullish, with worries that the president’s capacity for political self-sabotage, the Democrats’ fund-raising advantage and the anti-Trump intensity propelling the left will make it difficult to do much more than break even and protect its one-seat Senate majority.

But that Republicans are even discussing the prospect of gaining Senate seats, in the first midterm campaign of a president whose approval rating has never reached 50 percent, illustrates the wildly divergent electoral landscapes for the House and the Senate.

While the fight for control of the House is playing out mainly in the affluent and highly educated suburban districts that have been hotbeds of anti-Trump fervor, many of them on the coasts, the Senate campaign is taking place on much more Trump-friendly terrain. Six of the most competitive Senate races are in states he carried by double digits: Indiana, Missouri, Montana, North Dakota, Tennessee and West Virginia. (Democrats hold all of those seats except Tennessee’s.)