The Bay Area News Group reported that Crisp waived his no-trade rights as a 10-and-5 player (10 years in the big leagues and the last five with the same team) to allow the trade to take place.
The deal reportedly won't be announced until Wednesday, and it was not yet known what the A's would receive in return for Crisp.
Crisp was not in the Athletics' starting lineup Tuesday, and he left the Oakland clubhouse at Minute Maid Field in Houston about an hour before the start of the game.
Crisp, who will turn 37 in November, was limited to 44 games because of injuries in 2015, when he hit .175 with no homers. However, he has stayed healthy this season, hitting .234 wither 11 homers in 102 games. Crisp is a switch hitter but has played mostly against right-handed pitching.
Crisp leads major leagues with a .424 batting average with runners in scoring position.
He is scheduled to earn $11.75 million this year according to his contract with the A's. He has an option for $13 million that vests if he plays 130 games or has 550 plate appearances this season. The Indians only have 31 games left after Tuesday, and Crisp needs to play in 28 more games to reach 130. Crisp has 434 plate appearances so far.
In Cleveland, Crisp will compete with Abraham Almonte and Brandon Guyer for the fourth outfielder spot. Almonte is not eligible to play in the postseason after testing positive for boldenone in February.
The Indians led the American League Central by 4 1/2 games entering Tuesday's action, and they lost outfielder Michael Brantley to season-ending right biceps surgery on Aug. 15. Brantley hadn't played since May 9.
Crisp began his major league career with the Indians in 2002.