Conservative MPs across the board said they would block such a move, which requires two-thirds support in parliament.
"But if that happens people like me will just continue campaigning to get us out of the customs union", he added.
The proposal, which includes a bolted-on customs union arrangement, is building up steam after the main opposition Labour Party and the Scottish National Party backed it.
The Government has sought to play down any pressure coming from European leaders on the Border issue ahead of Mr Varadkar's visit to Paris to meet Mr Macron tomorrow and Ms Merkel's visit to Dublin on Thursday, insisting the engagements are about EU solidarity.
British lawmakers plan another round of votes Monday to see whether they can come to an agreement on a way for Britain to leave the bloc.
This will be the second round of indicative votes, which are created to find an alternative to the options they were presented with on Wednesday.
May has ruled out all the ideas under consideration.
Parliament will vote on different Brexit options today, and Mrs May could then try a last roll of the dice by bringing her deal back to a vote in the House as soon as tomorrow.
Defense Minister Tobias Ellwood urged fellow Conservative lawmakers to compromise to ensure an orderly Brexit.
Many Conservative MPs, including Boris Johnson and Jacob Rees-Mogg, changed sides to support her deal on Friday after May's promise to leave Downing Street if she won the vote.
If the government can't bridge the gap by April 12, Britain will crash out of the European Union without a plan for future relations, damaging its economy, undermining the country's unity and diminishing its stature in the world.
May's deal had already been rejected three times.
Slack rejected speculation that the government could take drastic action, such as asking Queen Elizabeth II to suspend Parliament or getting her to refuse to sign legislation.
"In any other circumstance, they would have rolled her by now, either formally or informally", said Tim Bale, a professor of politics at Queen Mary University of London.
If May chooses to ignore any alternative deal voted by Parliament, she risks a constitutional crisis and, potentially, a general election.
May has summoned ministers to a marathon Cabinet meeting on Tuesday to thrash out the options.
But if a customs union pledge was agreed by Parliament, it would likely be welcomed by the EU and would allow Britain to leave the bloc in an orderly fashion in the next few months.
However, another well-organised group of ministers, including Amber Rudd and David Gauke, are determined to avoid no deal and believe May should instead seek a cross-party consensus.
Last week, May said she would step down if her deal finally, somehow, gets over the finish line, thus allowing someone else to take the reins in the second phase of Brexit negotiations.
The Brexit impasse has alarmed businesses, who say the uncertainty has deterred investment and undermined economic growth.
So it could only happen if Mrs May was utterly determined to push it through - and she appears to have ruled out doing so - or if a summit of European Union leaders on April 10 decided not to give Britain a further year-long extension to sort itself out.
European Commission chief Jean-Claude Juncker said Monday that it was time for the British Parliament to spell out what it wanted on Brexit.
"Up to now, we know what the British parliament says no to, but we do not know what it says yes to", Juncker said.
But he added: "For the second half we are seeing softness in both the United Kingdom and Europe, which we believe comes from macroeconomic uncertainty and many unanswered questions surrounding Brexit which are together driving weaker customer demand".
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