According to Shams Charania of The Athletic and Stadium, the two sides haven't yet gained the "necessary traction" to get a deal done.
Parsons hasn't appeared in a game for the Grizzlies since Oct. 22 despite having been cleared to return to basketball activities since Dec. 21.
Memphis signed Parsons to a four-year, $94.5 million deal in 2016, but the move has not paid dividends.
The 30-year-old Parsons was limited to just 70 games combined in his first two seasons with the Grizzlies, and he has appeared in only three games this season.
During his time in Memphis, Parsons has averaged 7.0 points, 2.4 rebounds and 1.7 assists per game.
Parsons averaged better than 14 points per game during his previous stints with both the Houston Rockets and Dallas Mavericks.
While Parsons has just one year remaining on his contract beyond 2018-19, it is difficult to figure why the Hawks would want to acquire him.
Atlanta is rebuilding, and Parsons isn't particularly young.
Also, the Hawks don't have much in terms of big, long-term contracts, so they can't dump an albatross of a deal on the Grizzlies, either.
Parsons may still have some value in the NBA as a role player, but at his current price tag, it is difficult to envision a team willingly taking him off Memphis' hands.
Regardless of that, Parsons' time in Memphis seems to effectively be over since he can't crack the rotation of a team that has dropped to 14th place in the Western Conference at 19-28.