With the United Kingdom due to formally quit the European Union on March 29 next year, and three months needed to ratify the plans, a squeezed timetable carried the risk of a "no-deal by accident" scenario, ministers were told.
The Prime Minister today stressed to her Cabinet the importance of reaching an agreement this month but said this must "not be done at any cost".
Mr Johnson dismissed suggestions that the arrangement would lead the way to a Canada-style trade deal, arguing that the joint UK/EU political declaration on the future relationship due to be released alongside the withdrawal agreement will be "worthless".
However, a Downing Street spokesman said the intention of both the United Kingdom and the Republic is that the backstop should only be temporary.
But expectations are rising that United Kingdom negotiator Olly Robbins will be pressing hard to finalise a deal in Brussels this week, to set the scene for a special Brexit summit later in the month to secure the approval of the leaders of the 27 remaining member states.
"While the United Kingdom should aim to conclude the Withdrawal Agreement as soon as possible, this would not be done at any cost".
Her Cabinet is to hold talks over the plan early this week, with a potential European Union summit penciled in for the end of November, if they agree enough progress has been made.
"The prime minister said that, once agreement was reached on a withdrawal agreement, it remains the case that nothing is agreed until everything is agreed and it will be subject to securing an acceptable full future framework".
"There is still a real point of divergence on the way of guaranteeing peace in Ireland, that there are no borders in Ireland, while protecting the integrity of the single market".
In Ireland, Varadkar's openness to considering proposals for a review of the backstop, something that was communicated during a phone call with May on Monday, has been considered in some quarters to be a concession, even though he stated that "the outcome of any such review could not involve a unilateral decision to end the backstop".
But he made clear that he would not accept an arrangement which gave the United Kingdom unilateral powers to ditch the customs union without the agreement of Brussels.
A senior DUP MP has said the United Kingdom is heading for a no-deal Brexit.
However, the bloc has rejected United Kingdom attempts to place a time limit on any backstop arrangement, while Brexiteers have been pushing for the United Kingdom to have the unilateral right to exit a backstop arrangement and force a "no-deal" departure from the EU.
Looks like we're heading for no deal.
But allies of the PM were cheered that Cox had warned his colleagues that they risked a no-deal outcome if they insisted on a hardline plan to allow the United Kingdom to unilaterally pull out of any insurance policy on Ulster. "Given that neither was on the ballot in 2016, we believe the ultimate choice should be handed back to the public with a People's Vote".