The Lakers also get the Cavaliers' protected first-round pick in 2018.
The Lakers have been trying for months to clear future salaries to go after James, if he opts out of his contract, and George in free agency this summer. Getting the first-round pick gives the Lakers another asset to use to clear additional salary in the future.
Both Thomas ($6.3 million) and Frye ($7.4 million) are on expiring contracts. The Lakers now project to have $46.9 million in cap room this summer. The cap space will increase to $69 million if they do not bring back Julius Randle and stretch the $37 million owed to Luol Deng over five years.
Clarkson is under contract through 2019-20, and Nance's deal runs through 2018-19. But Cleveland picks up luxury tax relief with this trade this season.
Lakers coach Luke Walton said Thursday he plans to play Thomas and rookie Lonzo Ball in the backcourt together, even though they are both point guards.
Boston traded Thomas to the Cavaliers in August in the deal for Kyrie Irving, who had said he wanted out of Cleveland.
Thomas was rehabbing from a hip injury and ended up playing only 15 games for the Cavs. The two-time All-Star averaged 14.7 points, but Cleveland went 7-8 in those games.
"He's worked too hard to get back, and he's a ball-dominant player," Thomas' agent, Aaron Goodwin, told cleveland.com. "It's LeBron's ball, and this clearly wasn't working. [Cavs GM] Koby [Altman] and I have had enough conversations where it was clear, with the way the system was going, it wasn't beneficial for either party. This is a good opportunity for Isaiah."
The trade makes the Cavaliers younger, as both players they are getting are 25 years old.
Clarkson averages 14.5 points per game on 45 percent shooting. He had been the Lakers' sixth man, providing instant offense off the bench for much of the season.
Clarkson, who is owed $25.9 million over the next two seasons, averaged 21.2 points over an eight-game tear before scoring seven points in the Lakers' win over Phoenix on Tuesday. He leads the NBA with 274 points off the bench since Jan. 1, just ahead of the Clippers' Lou Williams (270) over that span, according to NBA Advanced Stats. Over the season, only Williams (813) has more bench points than Clarkson (732).
Nance averages 8.6 points and 6.8 rebounds per game. His father was a two-time All-Star for the Cavaliers.
He reflected on the trade in an Instagram post.
After three straight Finals appearances and a title in 2016, the Cavs are currently third in the East. The Cavaliers visit the Lakers on Sunday, March 11 (9 p.m. ET, ESPN).
ESPN's Ohm Youngmisuk contributed to this report.