The Prime Minister has been forced into the move in the face of threats from Brexiteers and Remainers to rebel.
Hardcore Tory Brexit-backers had intended to strike over clause 31 in the Taxation (Cross-Border Trade)Bill that would give the Government powers to implement a customs union with the EU.
And Remainers like Ken Clarke and Anna Soubry have been stoking a rebellion to keep Britain in the customs union with amendments to the legislation.
With speculation about which way Labour would go on the issue, the votes have now been kicked into the long grass in case of a defeat.
A source told The Times that both votes would now be pushed back and should not be expected “any time soon”.
They said: “This is about ducking any big decision for as long as possible.”
Eloise Todd, from the Remain campaign group Best for Britain, said: “Any delay to theses votes is a tacit admission that Theresa May does not have the numbers to see through her type of Brexit.
“She is in hock to more than 60 ideologically driven hard Brexiteers.
“This group would drive our country over a cliff for ideological purity.”
Shadow chancellor John McDonnell said today Labour was interested in creating a new customs union.
He said: “We are not supporting membership of the customs union but we are looking at a customs union.
“The reason we are saying a customs union is because we don’t want the same asymmetric relationship that Turkey have got.
“What we would want is to negotiate around our ability to influence the trade negotiations that would take place on behalf of us all, both ourselves and other European countries in terms of trade via a customs union.”
The news comes as Theresa May held crunch talks with Cabinet to thrash out the Government’s Brexit objectives.
The 11-member inner Cabinet Brexit committee are meeting at the Prime Minister’s country residence to decide a clear negotiating path as talks with Brussels step up a gear.
With the gathering expected to continue late into the evening, Mrs May is aiming to get clear Cabinet consensus on the details of Britain’s framework for a future relationship with the EU.
However, there was a setback to Mrs May’s plans on the eve of the meeting, as the European Commission released a document explicitly rejecting her “three baskets” approach to future regulatory co-operation.
The approach, set out in her speech in Florence in September, envisages different areas in which the UK would either continue with existing regulatory frameworks, operate its own separate rules largely mirroring those in the EU or diverge significantly from the Brussels regime.
But a set of slides setting out the commission’s negotiating position stated: “UK views on regulatory issues in the future relationship including ‘three basket approach’ are not compatible with the principles in the European Council guidelines.”
British attempts to “cherry-pick” EU rules threaten the integrity of the single market, while efforts to preserve UK influence over EU decision-making after Brexit “risk unsettling the EU ‘ecosystem”, the slides warned.