Discussions between the Mavericks and Sessions did not advance over the weekend, but Dallas and Houston appear to be at the top of the free-agent point guard's list because of the possibility of a starting role with either club.
Sessions would have made $4.55 million next season with the Los Angeles Lakers had he not opted out of the final year of his deal. The source said Sessions is seeking to begin negotiations north of that salary.
Although the 26-year-old initially had been attempting to secure a long-term deal, Sessions is willing to consider flexibility depending on the situation and details of an offer.
Dallas wants to keep its cap space clear for a run at a max-contract free-agent signing next summer and is interested in securing short-term deals.
The need for a starting point guard arose quickly last week when Jason Kidd, appearing set to agree to Dallas' offer of $9 million over three years, instead agreed to a similar offer with the New York Knicks.
Dallas has just seven players under contract -- and none are true point guards. Possibilities to play the position include combo guards Rodrigue Beaubois and Dominique Jones, plus draft pick Jared Cunningham.
Mavs free agent Delonte West remains a possibility to re-sign.
The Rockets are bare at the position after losing Goran Dragic in free agency to the Phoenix Suns and trading Kyle Lowry to the Toronto Raptors. Houston made a lucrative offer to Knicks restricted free agent Jeremy Lin, but New York is expected to match. It has 72 hours to do so once the NBA resumes business operations on Wednesday.
The Mavs have also been connected to Suns restricted free agent point guard Aaron Brooks, 27, who reportedly also has interest from the Rockets, the team that drafted him late in the first round in 2007.
Brooks won the NBA's Most Improved Player award in 2010, and played last season in China.