Brexiteers have raised fears that she could press for Britain and the EU to negotiate a fresh customs union to enable tariff-free trade to continue.
Bailey's comments come as talks between London and Brussels enter a crucial - and potentially fraught - second round, with the British prime minister coming under increasing pressure to clarify the future of Britain's relationship with Europe.
Mr Davis earlier stressed a "time-limited" transition period after 2019 would "lead to us being outside the customs union and indeed the single market in the longer term".
The draft legal text, which has yet to be approved by European Union governments, will be part of the withdrawal treaty that will regulate the terms of divorce between Britain and the European Union and the rules for the transition period after Brexit.
After days of vituperation about the customs union from both wings of the Conservative Party, Downing Street on Monday sought to dampen the row.
Britain, Davis said, wants a comprehensive free trade agreement and a customs agreement to make trade as frictionless as possible while being free to strike free trade deals with swiftly growing countries across the world.
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But Trade Secretary Liam Fox told Bloomberg: "It is very hard to see how being in a customs union is compatible with having an independent trade policy because we would therefore be dependent on what the EU negotiated in terms of its trading policies and we'd be following behind that".
According to United Kingdom media reports, Conservative party MPs in favour of a complete break from the EU have threatened to submit a "tsunami" of letters demanding a vote of no confidence unless May backs their demand of a complete exit from the EU customs union.
The desirability of belonging to a customs union with EU has been one of most divisive points between two sectors, most pro-European and Europhobe, in which ruling Conservative Party is divided.
He told Sky News that there was not a minute to lose in divorce talks. "The reality is, having alignment with the European Union on a customs arrangement is so far away from what the British public voted for, that that is what has to be delivered".
This comes ahead of the restarting of Brexit negotiations later this week in Brussels. This is what I think these critics are trying to do.
Members of the Cabinet Brexit sub-committee will meet on Wednesday and then again on Thursday as they seek to thrash out an agreement on thorny issues like customs arrangements with the remaining EU27. "We would need to add on a robust dispute resolution arrangement, but this could be done", he said. "It's not a matter of scaremongering to get negotiating advantage", he said.
Mrs May has made clear she will resist European Union proposals for these people to be entitled to settle permanently in the UK. People will be free to live and work here. That's the question MPs will be asking themselves when the Prime Minister finally shows her hand.