Ahead of a crucial week for the prime minister, the House of Commons leader admitted that “compromise will inevitably be needed” if the UK is to leave the EU with a deal.
Mr Johnson is preparing to brief his cabinet on the status of talks with the EU after a weekend of intense negotiations in Brussels.
The conference call is likely to reveal the likelihood of a deal being agreed in time for the 31 October deadline.
Even if he reaches agreement with the EU, however, Mr Johnson will need to win over Tory Eurosceptics and his Commons partners in the DUP in order go get the plan through parliament.
Writing in the Sunday Telegraph, Mr Rees-Mogg, a prominent Brexiteer, said the prime minister should be trusted because he had “dedicated his political career” to Brexit.
Of the prime minister, he said: “His charisma and intellect are undoubted, and his electoral appeal has been tested and proved in London.
“In the final stages of the Brexit negotiation, compromise will inevitably be needed, something even the staunchest Leavers recognise, albeit unwillingly – but, as a Leaver, Boris can be trusted. He wants to take back control and has dedicated his political career to this noble cause.”
And appearing on Sky News’s Sophy Ridge on Sunday, he added: “It is ultimately a question of trust about the direction of where we are going.
“I trust Boris Johnson to ensure the relationship the United Kingdom has with the European Union is one where we are not a vassal state.”
He insisted the UK would leave the EU on 31 October even if more time is required to enshrine a deal in law, saying: “If we get the deal and we have a vote possibly next Saturday and that is agreed, getting it through in legislation won’t be that problematic.
“The prime minister has been absolutely clear we are leaving on October 31 – this is of great importance.”
Eleven Tory peers have threatened to quit the party if Mr Johnson pushes for a no-deal Brexit.
The group, which includes former cabinet ministers, wrote in the Sunday Times: “We are particularly concerned that if the knife-edge negotiations fail or parliament blocks a deal, the Tory party may consider fighting a general election with a manifesto proposing a no deal, trying to pit ‘the people’ against parliament.
“Should this happen, the prime minister is likely to find himself with a much smaller parliamentary party in both houses – and significantly reduced support from previously loyal Tory voters.”
Mr Johnson is expected to hold a House of Commons vote next Saturday on any deal he brings back from Brussels. If no agreement is approved, the law says he must seek another Brexit delay from the EU.
According to reports, some former Conservative MPs will try to force him to request an extension even if he secures a deal. Former Tory rebels including Philip Hammond and Dominic Grieve are understood to believe that MPs should be given more time to scrutinise the proposed agreement, rather than it being rushed through parliament in a matter of days.
Mr Grieve told the Mail on Sunday: “He’s going to have to extend. I cannot see how he would be justified in trying to force through a major piece of constitutional legislation, the withdrawal agreement act, in seven days. It’s improper.”