CHICAGO — Senator Mark S. Kirk of Illinois, among the nation’s most imperiled Republican incumbents, sat before a small audience in the basement of a public library here recently, intent on proving his bipartisan bona fides — perhaps his only hope of defusing the wrath of voters in a state with little love for Donald J. Trump.

“I’m happy to be here with my bipartisan partner in crime,” Mr. Kirk said, referring to Senator Richard J. Durbin, the No. 2 Democrat, who was sitting two seats away and cringed mildly at the word choice. “Not crime,” Mr. Durbin said, shaking his head at a man he would like to see ousted from office.

The lack of enthusiasm was hardly a surprise. Mr. Kirk’s challenger, Representative Tammy Duckworth, is a protégée of Mr. Durbin’s — an Iraq war veteran who lost her legs when the Black Hawk helicopter she was co-piloting was shot down in 2004, and who got to know Mr. Durbin while she was recovering at Walter Reed Army Medical Center.

Mr. Kirk is among at least a half-dozen Republican incumbents in the Senate who find themselves in a most unusual electoral vise. They want to put distance between themselves and Mr. Trump, but in doing so, they risk alienating base voters in their states whom they need to win re-election.