The civil servant will reportedly leave his role shortly after briefing the new prime minister, who is expected to take office on 24 July.
The Independent understands that Mr Robbins is keen to move on, having remained as chief negotiator for eight months longer than originally intended.
It is believed that he may be considering a job in the City.
“I didn’t agree with the approach that we took in a number of areas of the Brexit negotiations,” he said.
“I was a loyal cabinet minister – I think it’s very important the prime minister has a loyal foreign secretary – but I wasn’t a believer in the backstop and I didn’t believe it would get through parliament, so I think this is a good moment to change our approach.”
His comments were met with cheers from Tory members.
“I’m not going to say something about a civil servant who I know works incredibly hard, even though there are issues I disagree with him on,” Mr Hunt added.
“But I do think now is a moment that we have to get this right, deliver a different deal and one that can actually get through parliament.”
Downing Street denied reports that the top aide had already resigned.
Mr Robbins, who was appointed by Theresa May, has been frequently criticised by Brexiteers over his role in negotiating the prime minister’s thrice rejected Brexit deal.
The 44-year-old is the prime minister’s Europe advisor based in the Cabinet Office.
Mr Hunt and Boris Johnson, who are competing to replace Theresa May as prime minister, are both expected to unveil new Brexit negotiating teams shortly.
The foreign secretary is believed to have recruited two Canadians with the aim of securing a Candian-style free trade deal, and Mr Johnson has reportedly drafted in Brexiteers including Jacob Rees-Mogg.
Additional reporting by agencies
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