Joe M. Rodgers, a hard-driving Nashville construction millionaire and a high-powered national fundraiser for President Reagan and the Republican Party, is expected to be named the next ambassador to France, White House and party sources said yesterday.

One source said the appointment was "a done deal," another described it as "a mortal lock" awaiting only Reagan's final sign-off. Evan Galbraith, the current ambassador, said on a French television newscast Tuesday night that Rodgers would be his successor.

A longtime supporter of Reagan, Rodgers, 51, was financial chairman of the Reagan-Bush Campaign Committee for the 1984 presidential campaign. In 1976, when most Tennessee Republicans lined up behind President Gerald R. Ford, Rodgers supported Reagan for the GOP nomination.

From 1978 to 1981 he was finance chairman of the Republican National Committee and a prime mover in the party's phenomenally successful fund-raising, credited with raising more than $100 million for the party since 1978 and substantially expanding the GOP "Eagles Club," contributors who give $10,000 or more.

"They had to tell people to stop sending money after he raised the legal limit [$16 million] for the primaries last year," one political observer said. In 1981, Reagan named him to the Intelligence Oversight Board.

Gov. Lamar Alexander of Tennessee noted that Rodgers has been "a very close and loyal friend of President Reagan and the French should be delighted to have someone that close to the president as their ambassador."

What the French will have is a prototypical American businessman -- deeply conservative, staunchly pro-American, pro-business and antilabor and an articulate exponent of his beliefs, a born-again Christian and a highly energetic, driven man -- some use the term "frenetic" -- who had a heart attack and bypass surgery eight years ago at the age of 43.

"He'll have an interesting time with the Socialist government of France," said Richard Lodge, the Tennessee state Democratic chairman and a Nashville lawyer who has represented Rodgers. "He's 100 percent All-American."

A native of Alabama, Rodgers' father was named Alabama state highway commissioner by Gov. George C. Wallace. He got a degree in engineering from Alabama University in 1956 and has three older brothers who also are engineers.

In 1966, he founded a construction company in Nashville that had total sales of $230,000. Ten years later sales were $140 million, but when Rodgers suffered his heart attack he sold 85 percent of the company to a Lebanese entrepreneur. He has since formed another construction company and two financial firms.

He also is a founder of Associated Builders and Contractors Inc., a group that promotes "merit shops," which to many are antilabor firms. A member of a nondenominational church, he has been national director of the Fellowship of Christian Athletes and associate chairman of the Layman's National Bible Committee.

He has also headed the Heart Association and Cancer Drive fund-raising campaigns and was commissioner general of the U.S. section of the 1982 World's Fair in Knoxville.