The Rockets are facing an NBA deadline of Sunday to either guarantee the deal for the season or waive Green.
The veteran guard will make $1.4 million for the season.
Houston plans to waive guard Bobby Brown on Friday, league sources said. He has a non-guaranteed contract.
The 31-year-old Green, who signed a free-agent deal one week ago, has had consecutive games of 27 and 29 points in the absence of the injured James Harden, and he has suddenly become an indispensable member of the Rockets. He had 29 points on 8-of-15 3-point shooting in a 124-114 loss to the Golden State Warriors on Thursday night.
The Rockets are expected to be without Harden for two to three more weeks because of a hamstring injury, league sources said, and Green has moved into the role of the team's sixth man in recent games. A native of Houston, Green has reveled in his return to his hometown team and has become an instant fan favorite.
The Milwaukee Bucks waived Green on the opening day of the regular season, and he had been working out in Houston until the Rockets called with an offer. Green has had NBA stops with Boston, Minnesota, Houston, Dallas, Brooklyn, Indiana, Phoenix and Miami, as well as stints overseas. He was the No. 18 overall pick in the 2005 NBA draft out of Gulf Shores Academy in Houston. Green won the NBA Slam Dunk Contest on All-Star Weekend in 2007.