Crawford is still owed $69.25 million through the 2017 season, part of the seven-year, $142 million contract he signed with the Boston Red Sox in 2010 after spending his first nine seasons with the Tampa Bay Rays.
Major League Baseball's non-waiver trade deadline is Thursday at 4 p.m. ET.
In August 2012, Crawford was part of an eight-player trade with the Dodgers that also brought pitcher Josh Beckett and first baseman Adrian Gonzalez to Los Angeles.
Trading Crawford would temporarily solve the logjam in the Dodgers' outfield. Matt Kemp, Yasiel Puig, Andre Ethier and Scott Van Slyke have been rotating in various outfield spots for most of the season.
Crawford, 32, is batting .242 with four home runs, 20 RBI in 55 games this season. He has spent time on the disabled list with an ankle injury.
- Scooby Axson