MPs could end up supporting another Brexit referendum if "none of the other options work", Tony Blair has said.
The ex-prime minister said there could be majority support for a new EU poll if Parliament ended up "gridlocked".
He urged Theresa May to "facilitate" the process by "running all options" by MPs first, including Norway and Canada-style alternatives as well as her deal.
But Labour frontbencher Angela Rayner says another referendum could increase division in the UK.
The shadow education secretary told the BBC's Question Time that holding a further Brexit vote would "undermine democracy".
"People made the decision and you can't keep going back saying, 'Would you like to answer it a different way?'"
And Hilary Benn, the Labour MP who chairs the Commons Brexit committee, has also distanced himself from calls within his party for another vote, telling BBC Radio 4's Political Thinking podcast that politicians have a "responsibility to give effect to the result of the last referendum".
But he told presenter Nick Robinson: "If the deal goes down… it may be that the prime minister decides, 'Well, I'm taking my deal to the country.'"
The government is opposed to any further referendum, saying the public made a clear choice when they voted in 2016 to leave by a margin of 51.9% to 48.1%.
Labour's position, which was approved by members at its party conference, is not to rule out any options if Parliament cannot agree a Brexit deal.
Shadow Brexit secretary Sir Keir Starmer said Labour's goal was to force a general election but if this was not possible, another referendum "had to be available" as an option.
The prime minister abandoned plans to hold a vote earlier this week on the withdrawal deal she has negotiated with the EU after admitting it would be heavily defeated.
She is currently in Brussels appealing to fellow EU leaders to soften their stance on the Irish backstop.
The PM insists her deal can pass if the EU is willing to give "political and legal assurances" on how the backstop - a contingency plan to ensure there is no hard border on the island of Ireland while the two sides settle a future trade deal - might come into force and how long it would last.
Mr Blair said he admired Mrs May's determination but suggested that, with so many MPs opposed to the backstop and other parts of the deal, this was becoming a weakness and she must realise she was "in a hole... and there is literally no point in carrying on digging".
Ahead of a speech in London later setting out the case for a "People's Vote", he said giving the final say to the people would become the "logical" outcome if every other option were to be exhausted.
"There will be a majority in Parliament, in the end, for a referendum if no other option of Brexit works," he told BBC Radio 4's Today programme.
"The real reason we should have another referendum is that we have had 30 months of negotiation and let's be clear, we are in crisis mode on this.
Will Labour’s leadership ever commit to a #PeoplesVote in line with conference policy?— Chris Leslie (@ChrisLeslieMP) December 14, 2018
How can they say a #PeoplesVote ‘undermines democracy’ but a general election doesn’t?
Of course the public have the right to think again!
Let’s get on with it now.https://t.co/GXVxyX2PAe
"The government is in a mess. Parliament cannot agree. Our knowledge has been vastly enlarged of what leaving the European Union will mean.
"If you look at all of this mess how can it be undemocratic to say to the British people, 'OK in light of all of this, do you want to proceed or do you want to stay?'"
Both the UK and EU are stepping up no-deal planning in case no agreement can be reached ahead of the UK's scheduled departure on 29 March.
Some Tory MPs, reportedly including some ministers, support a "managed no deal", in which the UK would reach agreements with the EU in key sectors, such as transport, and the UK would move to trade with the EU on World Trade Organization rules.
Cabinet Office Minister David Lidington said this was "not the government's policy" and ministers were focused on getting Commons backing for the deal in a vote now expected in mid-January.
Asked whether Parliament should hold a series of indicative votes on what the future course of action should be, he said the government first had to deliver on its promise that MPs would finish their debate on the proposed Brexit deal and vote on it.
Were the proposed deal to be defeated, he said the cabinet and prime minister would need to take stock and "go to Parliament accordingly".