Ms Cooper added she believes there is a majority across the country and Government in favour of a "close economic relationship", which means some version of single market participation - or as close as the United Kingdom can get to it.
The outcome could rest on the position of a handful of Brexit-supporting Labour MPs.
Varadkar made the comments as British Prime Minister Theresa May attempted to avoid an embarrassing defeat in parliament on the second day of debates over her post-Brexit trade plans.
They reassured anti-Brexit MPs that the government would accept some of their core demands to give parliament a meaningful say on the terms of Britain's European Union divorce, including - potentially - a new deadline for a deal to be agreed with Brussels that could make it hard for the government.
However, May is anxious about the prospect of a rebellion by pro-EU Conservative MPs who are determined to retain as numerous changes as possible.
MPs reversed a move to retain the UK's EEA links after it leaves the European Union next year, which had been backed by the House of Lords, by 327 votes to 126.
The Prime Minister was speaking as she confirmed that the Government will table a new amendment to her flagship EU Withdrawal Bill, setting out in more detail the terms of the "meaningful vote" promised to MPs on the final Brexit deal.
Here are the key amendments which MPs voted on Wednesday evening.
The fallout from Britain's referendum vote in 2016 to leave the European Union has reshaped politics, deepening divisions within its main parties and raising tensions between its four nations - England, Scotland, Wales and Northern Ireland.
This amendment had looked unsafe for the prime minister.
So far, the government has rejected all the other changes proposed by the Lords, including an attempt to remove the fixed Brexit date of March 29, 2019, from the bill.
One government official said: "It's not over yet".
A House of Commons showdown on customs between Tory Remainers and Brexiteers is now expected when a separate customs and trade bill is debated by MPs.
"It has got to be done in good faith, because without that, we will face a situation where in fact, firstly, the other place will put it back in and secondly the goodwill will be gone when it comes back to this house", leading rebel Dominic Grieve told lawmakers.
That amendment would compel the Government to prioritise the EEA deal, known as the Norway Model.
If that vote were to go against the government because a majority decided a better deal could be obtained, then parliament would be in chaos.
"The truth is I'm really anxious for my constituents, really anxious for the direction of travel that we are taking generally: I respect the decision in the referendum but we are closing off options of how we conduct future relationships in ways which are utterly damaging to ourselves". However, the party is divided on the issue, with a total of 89 Labour MPs choosing to defy Corbyn.
Among the Labour resignations was shadow cabinet office minister Laura Smith who voted against staying in the EEA. She was joined by Ged Killen, Ellie Reeves, Tonia Antoniazzi, Anna McMorrin and Rosie Duffield, who all resigned from their roles as Parliamentary Private Secretaries.
The Labour leader had ordered his MPs to abstain on a House of Lords amendment that demanded Britain's continued membership of the European Economic Area (EEA).
The 15 Labour EEA opponents included Frank Field, Caroline Flint and Dennis Skinner.
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