French President Emmanuel Macron said the agreement - 585 pages of legally binding text - "is the best accord possible. It is not re-negotiable".
Increasing numbers of government ministers have warned they will not accept the prospect of Britain leaving without a deal, which would immediately sever all ties with its largest trading partner.
"We are therefore asking you to work with your colleagues in Parliament urgently to find a solution that avoids the shock of a no-deal Brexit on 29 March and removes these risks for British consumers".
In order to cope with a no-deal Brexit, thousands of British companies are gearing up to move operations overseas if the United Kingdom crashes out of the European Union, the media reported.
The EU insists the Brexit agreement can not be reopened.
Ironically, rebellious hardline Euro-skeptics in May's ruling Conservative party, who were key in the heavy defeat of May's Brexit Withdrawal Agreement earlier this month, appear to be softening their opposition to her deal; while pro-EU Conservative rebels and middle-of the-roaders appear to be moving closer together in an alliance determined now to bury it for good.
Despite a last minute gamble aimed at buying off rebels in her Conservative Party, the prime minister will face a knife-edge battle to block a proposal that would hand the parliament the power to delay the process and prevent a no-deal divorce.
The imposition of tariffs and a collapse in sterling as a result of crashing out of the European Union could also lead to price hikes for shoppers, they warned.
Green Party legislator Caroline Lucas accused Mrs May of chasing, "heated-up fantasies that have already been rejected by the EU".
The rejection of May's European Union deal on Jan 15 left Britain hurtling towards the exit door with no plan.
Speaking to Sky News, Sir Graham said his move was an attempt to reach a compromise and with a "significant change" to the backstop there could be a majority for the PM's deal.
"We remain committed to ensuring all necessary legislation is in place for exit day on March 29, 2019, and it is important to stress we are confident of meeting that commitment", the spokeswoman said.
Indeed, EU officials were quick to reject any renegotiation.
Although Parliament is overwhelmingly opposed to May's Brexit deal, lawmakers are divided over what to do instead - whether to brace for a "no-deal" Brexit or to try and rule that out.
The EU has repeatedly said they would not renegotiate the agreed deal but they would consider extending article 50 process.
Tory former minister Nick Boles, a leading supporter of Ms Cooper's plan, told Today there was a "great deal of support among ministers and, indeed, Cabinet ministers" and they had been pressing for a free vote on the amendment. Some opposition Labour Party members sided with the government, anxious about being seen as obstructing Brexit.
The possible change of heart is being determined by their fear that pro-EU lawmakers are gaining in parliamentary strength.
The intervention is a fresh blow to Theresa May's hopes of salvaging her deal following the 230-vote drubbing two weeks ago.
The Prime Minister has reportedly told Tory MPs that the next meaningful vote motion will be laid on February 13, Sky News reports.
Robert Hazell, professor of government and the constitution at University College London, said the European Union was "pretty resolute in not being willing to reopen the negotiations unless the British government can come back with something more specific".
They warned: "We are extremely concerned that our customers will be among the first to experience the realities of a no-deal Brexit".