The power forward, who was a free agent earlier this year, did not have his contract renewed by the Clippers after a disappointing season. He has also been battling substance abuse issues and marital problems with his wife, Khloé Kardashian.
"He has been talking to them for weeks," a source close to the Kardashian family tells PEOPLE. "He's close to signing. They were definitely apprehensive but Lamar has shown huge improvement over the last few weeks and everyone around him believes he's sober."
Odom, 34, has been trying to get his life back on track since news of his drug addiction broke over the summer and he was arrested for DUI in August.
"He has a great support system. It's all up to him at this point. He's the one who can seriously f––– his life up," the source adds. "But he has the tools to stay sober and he's well-aware that one false move costs him his career."
Odom also struggled with trouble in his four-year marriage to Kardashian, 29, after infidelity rumors surfaced (which he denied) and the pair were living separately.
But the couple, who've been seen together recently, have been "going to therapy together," a second source tells PEOPLE.
Reporting by JENNIFER GARCIA and RAHA LEWIS