In a statement released Tuesday, Odom's aunt JaNean Mercer and the Odom family thanked doctors and nurses who cared for him at a Las Vegas hospital, asking supporters to keep praying for him.
"As Lamar begins a new chapter in his road to recovery, PLEASE continue to uplift him and the family in prayer. He continues to make miraculous progress, taking a few steps in Los Angeles. We couldn't be more overjoyed!" the statement said. "We realize Lamar's continued improvement will not be easy, however his unrelenting strength and faith in GOD will pull him through."
Odom was hospitalized in Las Vegas last week after he was found unconscious at the Love Ranch, a brothel in Nevada
where he'd spent several days.
A friend of Odom's told CNN that he has been transferred to a Los Angeles hospital. ESPN, citing sources
, reported that Odom was taken by medical helicopter on Monday night.
For days, those closest to him were tight-lipped about his condition.
Former college coach Jim Harrick said on CNN's "New Day" on Monday that the 35-year-old athlete has made "a little bit of progress" and has told his estranged wife, Khloe Kardashian, "I love you."
On Tuesday, Kardashian released a statement of her own, thanking doctors and nurses for helping Odom make "incredible strides."
"You can never be prepared for an experience like this, but without the outpouring of love and endless prayers that Lamar has received and the strength I was given from my loved ones, it would have been difficult to endure," she said. "Thank you for your continued support. God is great!!
Little is known about what left Odom unresponsive at the brothel
. Love Ranch owner Dennis Hof
recently said Odom was "in a good frame of mind" when he was taken to the brothel, and that he ate and slept well there.
"He picked a couple of girls, and he had the time of his life," Hof said.
In three days at the brothel, Odom spent $75,000, which guaranteed two women would "take care of any of his needs," Hof told CNN's sister network HLN.
Employees told 911 dispatchers that Odom used cocaine on Saturday -- his first day at the brothel -- though Hof said Saturday that any drug use happened when Odom was partying early that morning in Las Vegas but before he'd arrived at the Love Ranch.
"I'm going to protect my business, and I'm not going to let the world say that Lamar was doing a lot of drugs in my place," he told CNN's Michael Smerconish, saying his brothel has "zero tolerance" for illicit drugs. "Because it makes him look bad, and he wasn't doing that."
A woman said she also saw Odom take 10 herbal supplement tablets
over three days.
The brothel's employees said when they found him, Odom was lying in bed on his side with a pinkish fluid coming from his mouth and nose.
'More hopeful now'
Harrick said Monday that he'd been in touch every day with Kardashian, who had rushed to Odom's bedside.
The former NBA star is breathing on his own, Harrick said
"We're more hopeful now" that Odom will recover, Harrick said. "We've been on pins and needles for days."
Odom and Kardashian got married in 2009 and filed for divorce four years later, citing irreconcilable differences.
In addition to appearing on "Keeping Up With the Kardashians," the couple had their own reality show, "Khloe & Lamar." It lasted for two seasons and 20 episodes.
The two signed divorce papers this year, but the divorce has not been finalized, a source told CNN.
Trouble off the court
Odom won two NBA championships as a forward for the Los Angeles Lakers.
He played for four teams during his 14 seasons in the NBA and won the Sixth Man of the Year Award during the 2010-11 season.
Two years ago, he made headlines off the court when he was charged with drunk driving
. He pleaded no contest and received three years' probation.
Harrick, who coached Odom in college, said he believes the episode at the brothel is a "turning point" for Odom.
The athlete has a strong support system, Harrick said, and "prayers from everybody."