Mr Hancock was locked inside Downing Street for five hours with the Cabinet last night, as minsters finally approved the terms of the draft deal.
Speaking the morning after the deal was announced by Theresa May, Mr Hancock refused to confirm or deny his reported comments.
He told BBC Radio 4’s Today programme he would not go into what people had said behind closed doors, but said the prospect of no deal was ‘not pretty’ for healthcare.
He added: “We are working very hard to ensure that people have the unhindered flow and access to medicines that they need.
“There is a lot of work to do that but I hope that by this deal we can avoid a no-deal Brexit, we can avoid a second referendum and we can have a high-quality future with the EU, and that is what this deal provides.”
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Reports from the meeting suggested that tensions ran high, with Work and Pensions minister Esther McVey, who has since resigned, and Leader of the Commons Andrea Leadsom said to be ‘in tears’.
Mr Hancock said the discussions at the Cabinet meeting had been civil but ‘open and frank’, adding: ‘I didn’t see any tears.’
Mrs May faces a fight for her political survival today after four ministers, including Brexit Secretary Dominic Raab, and three ministerial aides resigned in protest.
Ms Leadsom and International Development Secretary Penny Mordaunt are also reported to be on the brink of quitting.
Prominent Brexiteer Jacob Rees-Mogg wrote to all Tory MPs urging them to reject the deal but Mr Hancock said MPs should back the PM’s Brexit plan because the alternatives to it are ‘ugly’.
He warned that failing to get the deal agreed with Brussels through Parliament would either lead to a no-deal Brexit or a second referendum and the risk of not leaving at all.
He told BBC News: “All MPs should vote for it because this deal is in the national interest.
“The two alternatives are deeply unattractive and as people read the detail of it and look at the deal in the round, rather than the bits and pieces that have come out in the newspapers during the latter stages of the negotiations, anybody in any compromise negotiated document can pick out individual parts that they would prefer were written differently.”