Foreign Secretary Mr Hunt – a potential contender in any future leadership contest – urged Europe to understand that working with Britain had brought peace and prosperity to the bloc, which would be endangered by a “huge, messy, ugly, horrible divorce” over Brexit.
“I want to make this point – don’t underestimate Theresa May,” he said.
“I think what’s happening at the moment is that after that very difficult general election [last year], maybe some people in Britain didn’t appreciate the steel at the heart of that lady.
“We are led by someone very tough and very resilient and I think now people in Europe are beginning to realise the same. Underestimating Theresa May is one of the biggest mistakes anyone can make right now.”
Mr Hunt said it was time for the EU to engage seriously with Mrs May’s Chequers blueprint for Brexit, rebuffed by the bloc last week.
He said: “We cannot have a situation…where every time we come up with a proposal, instead of engaging with that they simply say, ‘I’m sorry that doesn’t work, come back with something else’.
“Negotiations require two parties to engage seriously. That hasn’t been happening and Britain is not going to keep coming back with more.
“We have put on the table some practical proposals, which mean that we can honour the spirit and letter of the [EU] referendum and reassure businesses that they can have the frictionless trade that they want. They’re sensible proposals, we now need some engagement from the EU.
“At this stage in the negotiations it was always going to be like this, there was always going to be a moment where everyone was looking into the abyss, but now what we need is calm heads to prevail.”
“We need people to look at the very practical proposals that the Prime Minister has put on the table and I think when they look at them they realise that actually this is a very sensible way to deal with all the different tensions, all the different trade-offs.”
Mr Hunt said EU leaders had not treated Mrs May with respect last week, but they should see that while she always treated people courteously, she had deep principles, adding: “Never mistake British politeness for British weakness.”
Earlier, Mr Hunt, who was in New York for the United Nations General Assembly, was asked if Mrs May had seen off a leadership challenge. The Foreign Secretary said: “She is very secure. People understand that she’s someone of enormous tenacity, resilience. She’s had a very, very difficult few years with the Brexit situation.
“But she has brought us forward and she has brought us to the point where a deal is possible, and now we just need to see it through.”
But Mrs May’s de facto Cabinet deputy appeared to cast doubt on how long she will be leader.
David Lidington’s carefully-crafted remarks made to The Spectator magazine suggested the Prime Minister knows her political days are numbered.
They could also be seen as a bid to head off any immediate coup by reassuring rivals that Mrs May does not expect to make good on previous vows to stay put.
Asked if he wanted Mrs May to lead the Tories into the next election, Mr Lidington replied: “She said to the party that she will remain leader for as long as they want her to and I think at the moment people are absolutely backing her.”
Pressed again, he said: “She will decide, in due course, what she wants to do. But now, she is focusing on the task in hand.”
There is speculation that Mrs May will step down next year once Britain has left the EU, leaving her successor to iron out remaining Brexit details.
Meanwhile, leading Brexiteer Penny Mordaunt last night became the third Cabinet minister to publicly refuse to endorse Mrs May’s Chequers plan, saying only that she “supports” the Prime Minister.
And in another blow to Mrs May, The Spectator said last night three of her most influential Cabinet ministers had joined a group of Brexiteers discussing how and when to force her to ditch her proposal in favour of the Canada free trade deal route.