Speaking on BBC One's Andrew Marr show, Ms Leadsom said that it fell to Labour to accept the existing customs "arrangement" in Theresa May's three-times-rejected deal, which she described as "an excellent proposal".
"It would be, I think, a suicide note of the Conservative Party if we had to fight the European elections", he told BBC Radio 4's Today.
Three days of cross-party talks ended on Friday without agreement but Chancellor Philip Hammond said there were "no red lines" in the meetings.
He is himself under pressure from his MPs to demand another referendum on any deal he reaches with the government, with 80 Labour MPs signing a letter saying a public vote should be the "bottom line" in the negotiations.
"But I think both sides are committed to working quite rigorously to compromise as much as possible so that we can provide that compromise Brexit deal that I think parliament desperately needs at the moment". "If we can not secure a majority among Conservative and DUP MPs we have no choice but to reach out across the House of Commons". But we should be open to listen to suggestions that others have made.
In a statement released late Saturday, May admitted Brexit could only be delivered with support from the opposition Labour Party. "As I say, we had great discussions and we went into a lot of technical details but so far we haven't seen anything from government that would suggest they are prepared to change any part of the deal going forward".
Shadow Brexit Secretary Sir Keir Starmer said the government was unwilling to change the text of the Political Declaration.
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"My answer to that is clear: we must deliver Brexit and to do so we must agree to a deal. I'm not an advocate for no deal, but it would not be almost as bad as many like to think it would be".
The longer this lasts, the greater the risk that the United Kingdom will not leave at all, the prime minister underlined.
The PM has written to European Council President Donald Tusk to ask for a Brexit extension until June 30, however Mr Tusk is expected to offer a flexible one year extension which can be broken if the United Kingdom agrees on a deal.
Pat asked the Brexiteer whether he thought a no-deal Brexit could be salvaged after being "betrayed" by Theresa May and her government.
As things stand the United Kingdom is still set to leave the European Union on April 12 with no-deal.
The leaders of European Union member countries are due to meet in Brussels Wednesday - two days before the April 12 deadline - to consider Britain's request for a second extension.
Any extension would require unanimous approval from the other European Union countries, all tired of Britain's Brexit indecision, and could come with conditions.