Ben and Jerry's also urged ministers to talk to “expert organisations” including charity Refugee Action.
However, rather than take the brand's input on board, Mr Cleverly accused them of "virtue signalling".
Can I have a large scoop of statistically inaccurate virtue signalling with my grossly overpriced ice cream please. https://t.co/rkZFN9EO5y— James Cleverly (@JamesCleverly)August 11, 2020
He wrote: "Can I have a large scoop of statistically inaccurate virtue signalling with my grossly overpriced ice cream please?"
The MP's comments came after the ice cream maker tagged Ms Patel in a thread which began: "Hey @PritiPatel we think the real crisis is our lack of humanity for people fleeing war, climate change and torture. We pulled together a thread for you..
“People wouldn’t make dangerous journeys if they had any other choice.
“The UK hasn’t resettled any refugees since March, but wars and violence continue. What we need is more safe and legal routes.
“People cannot be illegal.”
The thread ends: “Let’s remember we’re all human and have the same rights to life regardless of the country we happen to have been born in.
“And once more for the back: PEOPLE CANNOT BE ILLEGAL.”
Immigration minister Chris Philp also hit back at Ben and Jerry's, telling the creators of Chocolate Fudge Brownie and Baked Alaska to “stick to ice cream”
Mr Philp told the ice cream company: “They’re ‘fleeing’ France, which is safe, civilised & has a good asylum system.
“Last year the UK made 20,000 asylum grants. We are the only G7 country to meet the 0.7% aid target & have run the largest refugee resettlement scheme in Europe over the last 5 years.
“Stick to ice cream.”
At least 12 migrants arrived at the port of Dover in Kent on Wednesday morning aboard Border Force boat Eagle.
They were helped ashore by officials in masks and fluorescent jackets and led up a gangway.
Border Force vessels continue to be active in the English Channel with more small boats expected later on Wednesday.
Lawyers representing asylum seekers who arrived in the UK by crossing the Channel on small boats have launched legal action to halt their deportation, which is due to take place on Wednesday.