Lawson, in his first season with the Rockets, has struggled the entire season with his play and has fallen behind veteran Jason Terry in the rotation in crunch time.
Rockets officials would like to keep Lawson, but his declining play is making it difficult for them to justify retaining him on the roster as they contend for the playoffs. Houston is eighth in the Western Conference standings, just a half-game ahead of Utah for the final playoff spot entering Friday night.
Rockets officials met with Thornton on Friday to discuss his prospects with the team. Houston wanted to make sure that if a move was made, Thornton still could be eligible to play for a playoff-bound team; sides must agree to a deal by Tuesday's midnight deadline for that to happen.
Lawson, meanwhile, is averaging career-lows in points per game (6.0) and field goal percentage (38.9) in 51 games. Since scoring 12 points and adding four assists in a Feb. 4 victory at Phoenix, Lawson has shot just 22.2 percent from the field and averaged 1.5 points and one assist in six games.
The Rockets acquired Lawson last summer, trading a lottery-protected 2016 first-round pick and four backup players to the Denver Nuggets. He was supposed to provide another ball handler to take pressure off leading scorer James Harden.
After coach Kevin McHale was fired 11 games into the season, interim coach J.B. Bickerstaff benched Lawson in favor of Patrick Beverley, who hasn't given up the job.
Lawson's off-the-court issues during his final season in Denver -- he had two DWI arrests -- have resulted in two league-mandated suspensions this season.
While he hasn't had any such issues with the Rockets this season, Lawson's play has been another matter. He played five minutes during Thursday's comeback win over Portland, missing the entire second half as Bickerstaff went with Beverley and Terry at the point guard spots.
"I want to see Ty succeed," Bickerstaff said before Thursday's game. "That's the only thing that's important to us, is seeing him play well, seeing him succeed, seeing him help the team. I know what's in there. We all know what's in there. I spend my time trying to figure out how to get it out. Do whatever it needs to be done to get it out."