The EU should show "common sense" and agree to make changes to the Brexit withdrawal deal, Boris Johnson has told the BBC.
The prime minister said there was "bags of time" for the EU to compromise on the Irish border backstop plan before the Brexit deadline of 31 October.
He also warned MPs not to oppose Brexit, and to respect the 2016 referendum result.
The EU has said repeatedly the backstop arrangements cannot be changed.
Mr Johnson said the policy - designed to guarantee there will not be a hard Irish border after Brexit - would turn the UK into a "satellite state" of the EU if it came into effect.
Speaking during a visit to Oxfordshire, he said the arrangements would "keep us locked in EU regulatory orbit, locked in the EU trading system, unable to control our own laws".
"We need change on that, once we get change on that I think we're at the races, and I think there's a good deal to be done."
Theresa May's deal has been rejected three times by MPs and as things stand, the UK will leave the EU on 31 October whether it has agreed a new one or not.
Many of those who voted against the deal had concerns over the backstop, which if implemented, would see Northern Ireland staying aligned to some rules of the EU single market.
It would also see the UK stay in a single customs territory with the EU, and align with current and future EU rules on competition and state aid.
These arrangements would apply unless and until both the EU and UK agreed they were no longer necessary.
No confidence vote
The prime minister's comments come amid growing speculation that opposition MPs will table a vote of no confidence in the government in a move to try and stop the UK leaving the EU without a deal.
Labour leader Jeremy Corbyn has said Labour will table a no-confidence vote at an "appropriate" time, once MPs return from the summer break in September.
According to the Sunday Telegraph, Mr Johnson's senior advisor at No 10, Dominic Cummings, has told MPs even losing such a vote could not stop the PM taking the UK out of the EU on 31 October.
He reportedly said Mr Johnson could call an election for after the deadline, with Brexit taking place in the meantime.
On Wednesday, shadow chancellor John McDonnell said he would expect Mr Johnson to step aside if he lost a no-confidence vote.
"No prime minister can sustain themselves in office when they no longer have a majority in the Commons", he said, speaking at the Edinburgh Festival.
"I don't want to drag the Queen into this, but I would be sending Jeremy Corbyn in a cab to Buckingham Palace to say 'we're taking over'."
Asked whether he would resign in those circumstances, Mr Johnson said MPs should ensure Brexit takes place by the end of October.
"What MPs should do and what they've already voted to do, when triggering Article 50 and reconfirmed several times, is honour the mandate of the people", he said.
On Tuesday cabinet minister Michael Gove accused the EU of "refusing to negotiate with the UK" over a new Brexit deal.
The European Commission said it was willing to hold further talks, "should the UK wish to clarify its position".