The BBC Breakfast presenter, 46, said she had since removed the likes and that they did not represent the views of the broadcaster.
Ending the interview, Stayt said: “I think your flag is not up to standard size, Government interview measurements.
“I think it’s just a little bit small, but that’s your department really. It’s just a thought.”
Mr Jenrick, who was speaking via video call from Westminster, did not respond.
When the camera returned to the studio, Munchetty was seen attempting to stifle her laugher.
She added: “There’s always a flag. They had the picture of the Queen though. In the Westminster office I am assuming.”
Munchetty later apologised for liking a series of tweets that referenced their interview and the flag.
She wrote on Twitter: “I ‘liked’ tweets today that were offensive in nature about the use of the British flag as a backdrop in a government interview this morning. I have since removed these ‘likes’.
“This do not represent the views of me or the BBC. I apologise for any offence taken.”
The BBC declined to comment further.
The corporation initially ruled that the presenter had breached editorial guidelines before then director-general Lord Tony Hall reversed the decision.