"Lamar today again has had moments when he is awake and more responsive," the source says. "He is still in the ICU, is receiving oxygen and sleeps a lot. He is not talking, but seems to understand when Khloé and the medical staff speak to him. He will have another head MRI in the next few days to check for any damage that could have been caused by the strokes.
"Khloé still hasn't left his side and looks very tired," the source adds. "As soon as Lamar wakes up, she rushes over to hold his hand. She sleeps on a pullout bed next to Lamar."
Early Sunday morning, Kim Kardashian West shared a message of encouragement for Odom, saying she and sister Kendall Jenner were able to make him smile.
"I have faith in you," she wrote. "I believe in the power of prayer and positive energy! Thank you all for your prayers! Lammy – I'm so happy Kendall & I could make you smile today! God is good!"
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A source previously told PEOPLE that Odom was conscious and "able to say hi to Khloé" on Friday afternoon, but that "his condition remains guarded."
The 35-year-old, who sources told PEOPLE on Thursday was breathing with the help of a ventilator at Sunrise Hospital, was found with both cocaine and opiates in his system.
A source close to the athlete's estranged wife Khloé Kardashian told PEOPLE that Odom's medical improvement has improved the spirits of his family and friends.
"The good news and Lamar's improvement really lifted their spirits," the source said. "They're all just praying it continues."