The Sunday Telegraph disclosed the Government will unlock billions of pounds into the economy if both sides fail to conclude a final agreement.
The spat comes at a sensitive time for Brexit negotiations and also for May's leadership.
With Britain refusing to give ground on the so-called divorce bill - and European Union leaders digging in on guaranteeing European Union citizens' rights - they are not expected to end the stalemate.
The British delegation at the fifth round of talks, starting in Brussels is led by David Davis, the EU exit minister, Europeans are represented by the main EU negotiator for Brexit, Michel Barnier.
The Customs Bill will create a new "stand alone" customs regime post-Brexit, regardless of whether any deal is agreed with the EU.
It is widely hoped that, following on from May's Florence speech and the generally encouraging response from some European Union leaders, enough progress can be made for the two sides to start discussing trade and transition, although expectations have been played down.
"What we don't do is run around advertising it demonstrably. Why?"
The simple fact is that half a loaf is always better than starving to death, although personally I think we should be aiming a tad higher than avoiding death.
May earlier this year warned EU Commission President Jean-Claude Juncker that she'd be a "bloody hard woman" to deal with in Brexit talks.
While the 27 governments have been united so far, that consensus may not hold if trade talks are postponed beyond December, according to the diplomats.
Mrs May has promised MPs a "take it or leave it" vote on any deal.
But what Ms May's problems inside her own Government indicate to many across the channel is that the country is so divided it may be unable to reach a deal.
It is the final set of talks before European Union leaders meet to decide if enough progress has been made to talk about post-Brexit relations with the UK. If the European Council has given the green light for talks to move onto the second phase, these will focus on trade relations after Brexit. The EU said "sufficient progress" is needed on these separation issues before it will allow the start of trade talks.
The jurisdiction of the European court is a inflammatory issue for some Brexit campaigners within May's Conservative Party.
61-year-old British PM Theresa May offered concessions in her speech in Florence but the pressure still hasn't detached from the negotiators to uplift the progress and diminish the uncertainty for businesses and citizens in the Britain and Europe.
Mr Ashworth reportedly said he was confused by the suspension: "The vote was not about disrupting Brexit and the negotiations". Asked what she would do about Johnson, who has repeatedly undermined her in recent weeks, she replied: "It has never been my style to hide from a challenge and I'm not going to start now".
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