WASHINGTON — Senator Lamar Alexander, Republican of Tennessee and one of the last bridges to bipartisanship in the Senate, announced on Monday that he would not seek re-election in 2020, citing a desire to leave the Senate “at the top of my game.”

“I’ve had my turn,” Mr. Alexander, 78, said in an interview. “Everything comes to an end sometime, and it is good to know when that should be.”

The departure of Mr. Alexander, a popular former two-term governor who twice unsuccessfully sought the Republican presidential nomination, will leave the polarized Senate without one of its few members with proven deal-making experience. His accomplishments included this year’s measure aimed at easing the opioid crisis and a second bill assuring more royalties to songwriters — a topic close to Mr. Alexander’s Nashville heart.

His decision to leave is more evidence that Washington has become less attractive to legislators interested in steering a middle course on seemingly intractable issues such as education and health care.