The Texans were one of the first teams to contact Leinart after he was released Saturday and have been among the most interested teams, according to the sources.
The Cardinals parted ways with the former Heisman Trophy winner days after Leinart went public with his frustration over being replaced by Cleveland castoff Derek Anderson.
Leinart complained he had outplayed Anderson and that his problems with coach Ken Whisenhunt were "probably away from football."
That led to a meeting with Whisenhunt the following day. Although he was unhappy that Leinart took his issues public, Whisenhunt praised how the quarterback has handled things.
"The one thing I want to make very clear is how professional Matt was about the whole situation," the coach said. "I was very impressed with his conversation with me. He thanked the organization, the ownership, for all the time and effort that they had invested in him. He felt like he had gotten better and he had learned a lot and it meant a lot to me to hear him say that."
Leinart played two series in the final preseason game on Thursday night.
Arizona thought it had a steal when Leinart, who led Southern California to two national championships and a third title game appearance, fell to the Cardinals at the No. 10 overall pick in 2006. He started 11 games as a rookie under then-coach Dennis Green, then the first five the following season under new coach Ken Whisenhunt. But the big left-hander then went down with a broken collarbone and Kurt Warner took over.
Warner finished his career with two spectacular seasons, leading Arizona to consecutive NFC West titles, an incredible run to the Super Bowl and a 51-45 overtime victory over Green Bay in a playoff thriller last season.
Leinart threw for 3,893 yards with 14 touchdowns and 20 interceptions with Arizona.
Information from The Associated Press contributed to this story.