Theresa May is due to make a statement on Brexit to MPs later, which will be followed by a debate on next steps.
MPs are then expected to vote on a series of alternatives to the prime minister’s Brexit deal.
Ministers were feeling “more positive” about being able to hold a third vote on the PM’s deal this week, sources told the BBC’s Laura Kuenssberg.
But Northern Ireland’s DUP has said its position has not changed and it will not be backing the deal.
Mrs May’s EU withdrawal agreement has been overwhelmingly rejected in the Commons twice.
She has said she would only bring her deal back for a third Commons vote if there was “sufficient support” for it – and she spent the weekend trying to persuade Brexiteer Tories to get behind it.
But many are thought likely to take their lead from the DUP, which has led objections to the Irish backstop clause.
Meanwhile, the EU has said all its preparation for an “increasingly likely” no-deal scenario on 12 April has been completed.
Later, MPs are expected to back a plan to carve out parliamentary time for a series of so-called indicative votes on alternatives to Mrs May’s deal.
As many as six other options, in addition to Mrs May’s deal, could be put to votes to see which are most popular.
Conservative MP Sir Oliver Letwin told BBC Radio 4’s Today programme that once Mrs May knew what it would take to get a majority vote, it would help her find “a way forward in principle”.
But International Trade Secretary Liam Fox told the BBC that Parliament may want to look at a series of Brexit options, but they cannot be binding on the government.
“I’m answerable to my voters not to the House of Commons,” he said.
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He told the Today programme there had to be an agreed deal by 11 April, otherwise the UK will have to take part in EU elections, which “would unleash a torrent of pent up frustration from voters”.
The indicative votes are a process for MPs to indicate which version of Brexit they might like if they don’t fancy the prime minister’s deal.
But there’s a clash in government over whether or not they should go into this process at all.
Parliament is going to do this anyway and the government has given a commitment for MPs to be able to have their say on a series of different ideas.
To be clear, it would not bind the government – even if there is one option that gets a clear preference from Parliament.
It would still have to get through the cabinet and it would still have to be workable for the Tory party.
That could then mean if Parliament puts down a marker to have a softer Brexit, Theresa May is stuck with the same problem she’s had all along: if she moves to something softer she might implode the Tory party.
Quite openly now, people in government are talking about something more dramatic as a way out.
Cryptically they call that a “democratic event”. What would we call that? An election.
Boris Johnson has described some of the suggested options – including a Norway-style close relationship with the EU – as “catastrophic” in an article in the Daily Telegraph.
Accusing Mrs May of “bottling” Brexit, the former foreign secretary said the only argument for backing what he called her “rotten deal” was if every other option was worse.
Meanwhile, Foreign Office Minister Mark Field said he would support revoking Article 50 – the two year process for leaving the EU – if it became an option in the event Mrs May’s deal was defeated and free votes granted for indicative votes.
Labour MP Wes Streeting said he believed there to be a “genuine desire” to find a way through the deadlock “but the prime minister has to set Parliament free”.
And Labour MP Peter Kyle said “what the country really wants” is for “grown ups to get a grip on this and show a creative and a solid way out of the madness”.
The European Commission said it had completed its no-deal preparations, which it said would cause “significant disruption for citizens and businesses” and “significant delays” at borders.
“In such a scenario, the UK’s relations with the EU would be governed by general international public law, including rules of the World Trade Organisation,” a statement said.
What’s happening this week?
Monday: MPs will debate the Brexit next steps and a number of amendments – possible alternatives – to the government plan will be put to a vote. The most important of these is the indicative votes plan.
Tuesday: Theresa May could bring her withdrawal deal back for the so-called third meaningful vote. But the government says it won’t do that unless it’s sure it has enough support to win.
Wednesday: This is when indicative votes would be held – we don’t know yet whether MPs will be free to vote how they want or be directed along party lines. The chances of any genuine cross-party consensus being achieved are not high.
Thursday: A second possible opportunity for meaningful vote three. The prime minister may hope that Brexiteers will finally decide to throw their weight behind her deal because indicative votes have shown that otherwise the UK could be heading for the sort of softer Brexit they would hate.
Friday: This is still written into law as the day the UK leaves the EU, but the PM has said she will try to change that this week. If she succeeds, the earliest Brexit will happen is 12 April.
On Sunday, amid reports of a plot to replace Mrs May with a caretaker prime minister, two cabinet ministers touted as potential successors said they fully backed the PM.
As senior figures dismissed talk of a “coup”, Mrs May summoned leading opponents of her deal to Chequers, her country retreat, to assess whether there was enough support for it to bring it back to the Commons this week.
But after lengthy talks with prominent Brexiteers – including Boris Johnson, Jacob Rees-Mogg and Iain Duncan Smith – there was little sign of an immediate breakthrough.
Government vs Parliament
In normal times, the government runs the country and Parliament – comprising all the MPs and Lords who are not members of the government – is there to monitor and scrutinise the way they are running things.
The government cannot make new laws or raise taxes without Parliament’s agreement. And Parliament can challenge or block decisions many of the by government ministers.
But ultimately it is the elected government that calls the shots – partly because it controls what gets debated in the Commons.
A group of MPs is now bidding to take over the Commons timetable on one day this week, so it can hold votes on alternatives to the government’s Brexit plans.
The government does not have to abide by the outcome of these votes, but it is trying to work with the MPs to avoid a showdown that could further undermine its already weakened authority.
Brexit: PM under pressure over Commons Brexit votes}