The 30-year-old Crawford made his request three weeks ago, but with the Hawks focused on re-signing All-Star Joe Johnson and other free agents, talks have not progressed to the serious stage.
The 6-foot-5 shooting guard is in the final year of a contract that will pay him $10 million next season.
If he doesn't sign an extension, he'll become one of the league's top free agents next summer. Crawford is happy in Atlanta and hopes to stay there for the remainder of his career.
The 10-year veteran is coming off his finest season, averaging 18 points on career-best 45 percent shooting in being named the NBA's top reserve. In helping the Hawks to a 53-29 record, their best mark in 13 years, Crawford reached the playoffs for the first time in his career.
Crawford was the Hawks' second-leading scorer behind Johnson during both the regular season and the playoffs.
During their disappointing second-round series with Orlando, in which they were swept in four games, Crawford led Atlanta with 17 points per game.
Like many potential free agents, Crawford would like to sign an extension before the league's current collective bargaining agreement expires after next season.
There is widespread belief around the league that there will be a lockout next summer, and many executives agree the next collective bargaining agreement will not be as favorable toward players.
Chris Broussard in a senior NBA writer for ESPN The Magazine.