"She wants to be there so that whenever he opens his eyes, he sees her," the source says.
Kardashian, 31, "is happy he is off the ventilator and is breathing on his own, but she is still scared and still concerned," a family source also tells PEOPLE. "He's woken up and his heart is stronger, but there is still concern about his organs."
Both sources confirm that Odom, 35, is no longer on a ventilator. "Things are looking brighter," the family source says.
But Odom has not been texting, the source says.
He has been able to talk, though it's unclear how much, the family source says: "I think it's minimal."
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Cocaine and opiates were found in his system and he had suffered multiple strokes, PEOPLE confirmed.
"His organs still aren't that strong," the family source says, adding that "they don't know the damage."
While some of the Kardashian family members have left Odom's Las Vegas hospital – sisters Kendall Jenner and Kim Kardashian West and their mom, Kris Jenner, visited this weekend – Khloé plans to remain by his side.
"Lamar still has a way to go," the source says. "It's promising, but he is not home-free."