Multiple media outlets also are reporting a Smith deal with Houston.
The veteran forward, let go Monday by the Detroit Pistons less than halfway into his four-year, $54 million contract, cleared waivers Wednesday.
The Rockets next take the court against the host Grizzlies at 8:30 p.m. Friday.
Houston is three games behind Golden State in the Western Conference standings and just a half-game behind Memphis for the 3-seed.
As ESPN.com's Marc Stein reported late Tuesday, Houston ?assured Smith that he would start if he joined the Rockets, which essentially clinched his decision.
The Rockets also had their biannual exception, valued at $2.1 million, to offer Smith, who is 29.
Sources said Wednesday that Smith made it known that he would only consider teams that would guarantee him a starting spot, which essentially made it a choice between Houston and Miami.
Houston center Dwight Howard also could have played a factor in Smith's decision. Howard, one of Smith's closest friends, played AAU basketball with Smith in Atlanta
Smith is averaging 13.1 points, 7.2 rebounds and 4.7 assists per game this season.
Smith will be playing for coach Kevin McHale, who signed a three-year extension on Wednesday.
Chris Broussard, Marc Stein and Ramona Shelburne of ESPN.com contributed to this report.