Some media reports suggested that UK Chancellor Philip Hammond had angered some of his more pro-Brexit Cabinet colleagues with his concerns over plans to immediately restrict immigration from the EU, warning that this could harm the UK economy.
It came as leading Brexit-backer Boris Johnson was forced to defend himself over an unpublished article which showed him presenting arguments in favour of staying in the EU.
"If we weren't having lively debates in cabinet, you would be saying what's happened to Cabinet government, why aren't you going through incredibly thoroughly, all the different arguments to make sure we end up with the right decision," Hunt told BBC.
"The British people changed the history of our country on June 23 and the Cabinet is absolutely united that we must respect that decision. But we need to then go through very thoroughly and carefully, all the different options," he said.
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An unnamed cabinet source told 'The Daily Telegraph' that Hammond was "overly influenced by his Treasury officials who think it is a catastrophe that Britain voted to leave the EU".
Another source told the paper: "He is arguing from a very Treasury point of view. He is arguing like an accountant seeing the risk of everything rather than the opportunity."
According to the 'Guardian', differences between ministers are likely to surface most strongly on the Brexit Committee, whose 12 permanent members are split evenly between ministers who supported the leave and remain campaigns before the June 23 referendum.
Yesterday, an unpublished column by Johnson, now UK foreign secretary, created a stir as it argued for continued EU membership.
The senior Cabinet minister, however, made light of the article, saying it was a result of the fact that he was "wrestling" with his stand on Brexit.
The debates and divisions over the exact nature of Brexit are expected to intensify in the lead up to the deadline.