Mr Allen, 67, drew criticism after telling LBC he was “bored” with the 19-year-old’s performances on the show.
He added her physical appearance was probably to do with “her dad’s cooking”.
Ms Ramsay, daughter of celebrity chef Gordon, hit back on social media shortly afterwards, saying she “won’t tolerate people that think it’s OK to publicly comment and scrutinise anyone’s weight and appearance”.
She added: “Please remember that words can hurt and at the end of the day I am only 19 and I’m so grateful for all the amazing opportunities I have been able to take part in and I understand that being in the public eye obviously comes with it’s own repercussions and I’ve been aware of this from a young age.”
Ofcom, the broadcasting watchdog, said it had received 840 complaints relating to Allen’s comments about Ramsay.
A spokeswoman said: “We are assessing the complaints against our broadcasting rules, but are yet to decide whether or not to investigate.”
On his radio show, Allen had read out a comment from a listener telling him the teenager is taking part in Celebrity MasterChef Australia.
He replied: “Is she? Well, she can’t blimming well dance, I’m bored with her already.
“She’s a chubby little thing, isn’t she? Have you noticed? Probably her dad’s cooking, I should imagine.”
His comments drew a furious reaction from celebrities including Love Island’s Dr Alex George, who said he was “horrified” by the remarks.
Dr Alex, 30, said in his letter a “discussion about a young person’s weight live on air is not acceptable”.
Mr Allen said he had contacted Ramsay last week to issue a private apology.