But Mr Johnson insisted that a separate codicil setting out amendments to the backstop plan would not be "good enough", indicating that he is holding out for changes to the text of the agreement itself.
Mr Barclay will later travel to Brussels for talks over dinner with the EU's chief negotiator Michel Barnier.
Mrs Leadsom said there was "no chance" Mrs May would adopt Mr Corbyn's "view of the world", adding: 'The Prime Minister has been absolutely clear we're leaving the EU, we're leaving the customs union, we're leaving the single market'.
"It's clear that Jeremy Corbyn... he's done a complete U-turn", Mr Johnson told BBC Radio 4's Today programme.
'So he wants a customs union and he is unclear as to whether that means he also wants an independent trade policy.
That is the insurance policy to ensure no hard border reforms on the UK's frontier with the European Union - between Northern Ireland and the Republic.
But they remain fundamentally divided over whether the United Kingdom should ask to be in a customs union.
"I think it must be pretty obvious that if you are going to have a time limit to the backstop, and I think that would be a very good thing, it's got to fall before the next election".
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In her letter, Mrs May said she wanted discussions between Tory and Labour teams to start considering "alternative arrangements" to the backstop contained within the withdrawal agreement.
She insisted her deal already met numerous conditions he had set.
That leaves May battling to persuade a reluctant European Union to look again at the Irish backstop - a fallback policy created to prevent the resurrection of a hard border in Ireland if talks to find a long-term trade arrangement fail.
The government has signed a trade deal with Switzerland, heralded as the most significant such agreement in the run-up to Brexit.
Cabinet will meet this morning, with Downing Street hoping it has seen off any immediate risk of resignations over the looming prospect of a no-deal Brexit by promising a further round of Brexit votes at the end of February, if Mrs May still hasn't achieved the renegotiated deal she is calling for from Brussels.
This is of huge economic importance to United Kingdom businesses so I'm delighted to be here in Bern today, ensuring continuity for 15,000 British exporters. Not only will this help to support jobs throughout the United Kingdom but it will also be a solid foundation for us to build an even stronger trading relationship with Switzerland as we leave the EU.
"By getting the changes we need to the backstop; by protecting and enhancing workers' rights and environmental protections; and by enhancing the role of Parliament in the next phase of negotiations I believe we can reach a deal that this House can support", May will say.
She has said she wants to bring a revised deal back to parliament for a vote "as soon as possible" but has not yet set a date for doing so.