In a written statement, reports CBS News Capitol Hill Correspondent Bob Fuss, Thompson says he simply "does not have the heart for another six year term," and hopes his supporters understand. Thompson, 59, completing his first full term, has had both a political and an acting career and first came to public attention working for the Senate Watergate committee.
He recently lost his 38-year old daughter, Elizabeth Thompson Panici, who died following a heart attack. Thompson, who has two other children, is divorced.
Thompson had announced his retirement this summer, then, after Sept. 11, reports CBS News Political Director Dottie Lynch, decided that public service was a career that was worthwhile, and said he'd seek re-election.
Gore, who would have won the U.S. presidency in 2000 if he had carried his home state of Tennessee, said he would campaign for whoever gets the Senate Democratic nomination.
Thompson becomes the fourth Republican senator to step down, setting up a competitive race in Tennessee and complicating Republican efforts to try to gain the one seat they need to take back Senate control.
CBS News White House Correspondent Mark Knoller reports President Bush issued a written statement hailing Thompson, saying he has served the people of Tennessee with honor, distinction and class.
"He has worked tirelessly for Tennessee's interests, as well as for the national interest," Mr. Bush said. "While I will miss Fred's service in the Senate, I wish him all the best and will always call him a friend.'
Before being elected, Thompson acted in such films as "No Way Out" and played an admiral in the movie "The Hunt for Red October."
Former Tennessee Gov. Lamar Alexander, who failed in two bids for the Republican presidential nomination, is expected to announce his candidacy for the seat, Republican sources said.
On the Democratic side, Reps. Harold Ford, John Tanner and Bob Clement could decide to join former gubernatorial nominee John Jay Hooker in a battle for the party's nomination to replace Thompson, Democratic party sources say.
"The Tennessee Senate race...has just become an incredible opportunity for a Democratic pickup," said Sen. Patty Murray, D-Wash., who chairs the Senate Democrats' re-election efforts.
Democrats had not been able to find a strong candidate to run against Thompson, said Lynch, but now it's a race again.
Democratic sources said party leaders would contact Gore to see if he would run for the Senate, where he served from 1984 to 1992 before he was elected vice president.
"He obviously would be our strongest candidate," a Democratic strategist said.
But Gore released a statement rejecting the possibility even before Thompson announced his decision to retire from Congress.
"We have some outstanding Democratic leaders in Tennessee who I hope will be candidates," Gore said. "I will work hard to elect one of them to the Senate but I will not be a candidate for the Senate myself."
Gore, who has not yet said whether he will make another bid for the White House, recently established a political action committee that he intends to use to help fellow Democrats in this November's election.
The four Republican Senators who have announced they will not seek re-election are Phil Gramm of Texas, Jesse Helms of North Carolina and Strom Thurmond of South Carolina.