Republican leaders consider next year’s midterm elections a ripe opportunity to strengthen their party’s majority in the Senate, but GOP lawmakers are turning down challenges in states in which President Donald Trump defeated Democrat Hillary Clinton.

Rep. Ann Wagner, a Republican who represents the Second Congressional District in Missouri, said Monday that she won’t challenge her state’s senior U.S. senator, Claire McCaskill, a Democrat who is running for re-election in 2018. Mr. Trump won Missouri by 19 points last year....