"He is so nice and being as supportive as humanly possible given the situation," a source close to the Kardashians tells PEOPLE. "Lamar essentially has zero consistent family other than the Kardashians, and James understands that. The last thing he would want to do would be to add more stress to her plate."
Odom, 35, woke up from a coma Friday, four days after he was found unconscious in a Nevada brothel. The former basketball star is being treated at Sunrise Hospital in Las Vegas, and his estranged wife has not left since rushing there Tuesday.
Kardashian, 31, is still legally married to Odom even though they both signed divorce papers in July. The couple split in 2013, and she began dating Houston Rockets shooting guard Harden in June.
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"He never had any intention of going to Vegas. He supports her of course, but understands she needs space and to be with Lamar," another source tells PEOPLE.