Joan Rivers' daughter has released a statement updating the public on her mother's medical condition.

"On behalf of my mother and our family, we are extremely grateful for all the love and support we've received. At this time she does remain on life support," Melissa Rivers said in a statement obtained by E! News.

"I know my mother would be overwhelmed by the continued outpouring of kindness and I want to thank everyone for keeping us in their prayers."

Reports have surfaced that Melissa, who starred in the reality series Joan & Melissa: Joan Knows Best with her mother and also competed against Joan on The Celebrity Apprentice's second season in 2009, plans to sue the New York City endoscopy clinic where Joan underwent a minor throat procedure late last week.

A source close to the situation assured E! News the rumors "couldn't be further from the truth. Right now, [Melissa] is 100 percent focused on her mother's health. Any story suggesting otherwise is the stuff of fairies and elves -- it's fantasy."

Joan's family and friends are reportedly hopeful about her current condition, as everyone is "taking it day by day."

Joan, 81, remains at Mount Sinai Hospital in New York after going into cardiac and respiratory arrest during her August 28 procedure. It is unclear how long Joan went without breathing prior to being placed in a medically-induced coma.

Melissa, 46, flew from California to New York as soon as she heard Joan, who won The Celebrity Apprentice, had collapsed and was rushed to the hospital. She's apparently been at her mother's side since the incident.

E! issued a statement on the status of Fashion Police for which Joan, a famous comic, serves as host.

"We will not be producing a Fashion Police this week or next as we await Joan and Melissa's return in front and behind the camera," the network reportedly said. "E! will be airing two E! From Fashion Week specials that will deliver the latest news and information from the highly-anticipated event."