The BBC broadcaster and political commentator is currently undergoing physiotherapy after being admitted to hospital after having a stroke last week.
Marr's family thanked everyone who had sent them messages of goodwill, a BBC spokesperson said.
"Andrew Marr is continuing to make good progress and is working on his physiotherapy with his customary determination," the spokesperson stated.
"The Marr/Ashley family would like to thank everyone for the fantastic support they have received in terms of cards, emails, texts, tweets, flowers, etc.
"They would like to repeat their request for Andrew's privacy to be respected while he recovers."
It was previously announced that Marr's TV shows will be hosted by stand-ins while he recovers in hospital.
Andrew Marr has hosted his Sunday morning television programme since September 2005, having previously worked as editor of The Independent and political editor of BBC News.