The deadlock in parliament has also raised the prospect of Britain having to delay Brexit beyond March 29, something May is reluctant to do but which one official indicated could be an option if lawmakers refuse to pass her deal.
"We are heading in that direction but there is still more play in the day's ahead", he said.
Business, however, appears to have finally decided that derailing or watering down Brexit must finally take a back seat to simply biting the bullet and getting on with things one way or another, with the Institute of Directors complaining that a delay now would only "drag out uncertainty".
Shadow Brexit secretary Keir Starmer said May's delay was "the height of irresponsibility".
"I think that is counter-productive".
The House of Commons was set to debate and vote on May's supposedly altered deal next week as a little longer than a month is left before the March 29 Brexit date.
The Telegraph newspaper reported that May was considering a plan to delay Britain's for up to two months.
"We won't bring a meaningful vote to parliament this week".
May met with German Chancellor Angela Merkel and European Commission President Jean-Claude Juncker early Monday as she sought elusive changes to the U.K. -EU divorce agreement. You can't have it both ways; you cannot be in the government and against the government.
But the European Union has so far rebuffed May's attempts to reopen the withdrawal agreement.
A separate amendment proposed by Labour MP Yvette Cooper and Tory MP Oliver Letwin is also due to be debated on Wednesday, and would legally oblige the government to offer MPs the option of extending Article 50 - the mechanism by which the United Kingdom leaves the European Union - beyond 29 March.
The ongoing unrest over Britain's bid to depart the European Union marks the country's deepest political crisis in decades after almost 52 percent of Britons - more than 17 million people - voted to leave the European Union during the UK's June 2016 referendum. They will be returning to Brussels on Tuesday.
"MPs need to insert some rigour into this process", he said.
The immediate focus for traders is Wednesday when parliament votes on Brexit amendments.
She has so far stuck to her line that any extension would only defer a decision on Brexit rather than solving the impasse in parliament over whether to approve the deal.
He added: "The idea that we would put a time limit to that backstop but not be able to answer the question 'What happens after that time limit runs out and what replaces the backstop?' - then it wouldn't be a backstop at all".
Writing exclusively in The Yorkshire Post, the senior statesman of local government says economic and political stability will only be achieved if a way forward is found which "the British people deem fair and acceptable".