Boris Johnson has said he can see "a way forward" to reaching a deal with the EU in "all our interests" before Brexit is due to happen on 31 October.
But the prime minister warned the cabinet there was still a "significant amount of work" to do, as EU and UK officials continue to hold talks.
Parliament will meet on Saturday and vote on any deal achieved by Mr Johnson at a Brussels summit this week.
Labour said it would "wait and see" but would oppose anything "damaging".
The European Commission echoed the prime minister, saying: "A lot of work remains to be done."
Shadow business secretary Rebecca Long-Bailey told BBC One's Andrew Marr Show: "We don't think the Tories have moved too far on their deal."
SNP leader Nicola Sturgeon told the same programme: "We will not vote for the kind of deal specified by Boris Johnson."
Talks in Brussels between UK and EU officials - described as "intense technical discussions" - continued on Sunday and will re-start on Monday.
House of Commons Leader Jacob Rees-Mogg, a leading Brexiteer, told Sky News that "compromise" would be inevitable during negotiations.
He added: "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."
Mr Rees-Mogg also said he might have to "eat my words" and support a plan close to the one put forward by former Prime Minister Theresa May, which MPs rejected three times.
By Nick Eardley, BBC political correspondent
Is there going to be a deal then?
Forgive me a politician's answer, but the truth is nobody knows for sure. Not yet.
Both sides are being tight-lipped on the exact discussions happening behind closed doors in Brussels.
Indeed the cabinet was given very little detail about what exactly is being discussed.
Some might see that as a positive sign; nobody is going public on the concerns they have.
That doesn't mean they don't have them, but it suggests there is serious work going in to try to solve them.
I'm told Boris Johnson sounded genuinely confident in the cabinet conference call that a deal can be done.
Others in Westminster are filling up the coldest water they can find to pour all over reports a deal could be coming.
One opposition source told me they have war-gamed six potential outcomes for this mammoth political week.
They didn't give any of them more than a 50% chance.
Michel Barnier, the EU's chief Brexit negotiator, is expected to update ambassadors from the bloc's 27 member countries on Tuesday.
The summit in Brussels on Thursday and Friday is seen as the final chance to get a Brexit deal agreed ahead of the deadline of 23:00 GMT on 31 October.
A Downing Street spokesperson said: "The prime minister updated cabinet on the current progress being made in ongoing Brexit negotiations, reiterating that a pathway to a deal could be seen but that there is still a significant amount of work to get there and we must remain prepared to leave on 31 October."
The spokesman said Mr Johnson believed a deal could "respect the Good Friday Agreement", signed in 1998 in an effort to end the Troubles in Northern Ireland.
It could also "get rid of" the backstop - the plan to prevent the return of a hard border between Northern Ireland and the Irish Republic - which the government says threatens the future of the UK.
Liberal Democrat leader Jo Swinson told Sky News that any agreement reached by Mr Johnson should "be put to the public so they can have the final say".
But asked whether more MPs would be likely to support a deal, if the Commons first voted in favour of putting it to a referendum, Labour leader Jeremy Corbyn said: "I think many in Parliament, not necessarily Labour MPs - others - might be inclined to support it because they don't really agree with the deal.
"I would caution them on this."
Asked about Labour's stance, Home Secretary Priti Patel replied: "They are clearly playing politics. The British public want to ensure that we get Brexit done."
Mr Johnson's revised proposals - designed to avoid concerns about the backstop - were criticised by EU leaders at the start of last week.
However, on Thursday, Mr Johnson and Irish Prime Minister Leo Varadkar held talks and said they could "see a pathway to a possible deal".
The Benn Act, passed by Parliament last month, requires Mr Johnson to ask EU leaders for a delay to Brexit if a deal has not been reached and agreed to by MPs by 19 October.
The first Queen's Speech of Mr Johnson's premiership, delivered during the State Opening of Parliament on Monday, will see the government highlight its priorities, including on Brexit.
Timeline: What's happening ahead of Brexit deadline?
Monday 14 October - The Commons is due to return, and the government will use the Queen's Speech to set out its legislative agenda. The speech will then be debated by MPs throughout the week.
Thursday 17 October - Crucial two-day summit of EU leaders begins in Brussels. This is the last such meeting currently scheduled before the Brexit deadline.
Saturday 19 October - Special sitting of Parliament and the date by which the PM must ask the EU for another delay to Brexit under the Benn Act, if no Brexit deal has been approved by Parliament and they have not agreed to the UK leaving with no-deal.
Thursday 31 October - Date by which the UK is due to leave the EU, with or without a withdrawal agreement.