Boris Johnson has faced criticism after comparing French President Francois Hollande to a Second World War guard administering "punishment beatings".
Downing Street attempted to defend Mr Johnson over his use of colourful language, but he was condemned by political opponents.
Mr Johnson's remark came during a visit to India, when he was asked about a reported comment from one of Mr Hollande's aides, who said Britain should not expect a better trading relationship with Europe from outside the EU.
Mr Johnson responded: "If Monsieur Hollande wants to administer punishment beatings to anyone who chooses to escape, rather in the manner of some World War Two movie, then I don't think that's the way forward.
"It's not in the interests of our friends or our partners."
Mr Verhofstadt, the former Belgian prime minister who is the European Parliament's lead negotiator on Brexit, said the comments were "deeply unhelpful" and urged Mrs May to condemn them.
Mr Johnson's comments were denounced as "wild and inappropriate" by Labour, which said they would not help Britain negotiate a favourable Brexit deal with the other 27 EU nations.
The Prime Minister's official spokeswoman brushed off suggestions that the Foreign Secretary should apologise, describing his comments as a "theatrical comparison" and insisting he had not been comparing the French president to a Nazi.
Mrs May's spokeswoman dismissed the row as a "hyped-up media report", and said she was not aware of any complaint from the Elysee Palace. "He was making a point," she said. "He was in no way suggesting that anyone was a Nazi."
She said Mrs May and the Government were "focused on taking forward our plan for Brexit".