It is understood the UK Government and the European Union had been close to agreeing a deal guaranteeing "regulatory alignment" between Northern Ireland and the Republic in order to prevent a hard border after Brexit.
This would not be the same as "harmonisation" with the EU's rules, he said.
"The DUP tail is wagging the Tory dog", he said, in a reference to the deal struck between the parties to help the Conservatives survive key votes.
He said keeping Britain in the single market would be the way out of the "incredible mess" created by Prime Minister Theresa May.
Talks with European Commission president Jean-Claude Juncker broke up without agreement on Monday, although both sides said they were hopeful of getting a deal by the end of the week.
On Monday, it was widely rumoured that negotiators had managed to find a compromise where Northern Ireland would not diverge in regulations from the Republic of Ireland, which would negate the need for a hard border on the island.
The prime minister needs the support of the DUP, which is Northern Ireland's largest party and has 10 MPs at Westminster, because she does not have a majority to win votes in the House of Commons.
Mrs May is due to update ministers at a cabinet meeting later.
He said: "She flip-flops on this issue constantly and has done since before the European Union referendum".
There is little possibility of a fresh agreement over the border because Theresa May has taken two conflicting stances, in that she has said she does not want a hard border in Ireland but also that the United Kingdom will leave the customs union and single market.
Meanwhile, Scottish Conservative leader Ruth Davidson yesterday called for any regulatory alignment under Brexit to be applied on a UK-wide basis.
May said there had been no overall agreement on Monday.
He believed the DUP had been spoken to beforehand about what was being proposed, but that the precise wording had not been explained and the Northern Ireland party "clearly thought twice".
Mr Russell stressed it was membership of the single market that allowed the border between Northern Ireland and the Republic to "be completely porous".
Former top Foreign Office official Lord Ricketts said the row was "damaging" for Mrs May, adding that it was "pretty extraordinary that this wasn't all stitched up with the DUP beforehand". Either the prime minister must accept a softer Brexit that will alienate her Brexiteer backbenchers, or she will have to accept a hard border in Ireland that could be at odds with the Good Friday Agreement. The union said the recent announcement for Northern Ireland shows that it is possible to achieve what the FUW has been calling for since the EU referendum.
Downing Street has not responded to Mr Varadkar's claim.
"Huge ramifications for London if Theresa May has conceded that it's possible for part of the United Kingdom to remain within the single market and customs union after Brexit", Sadiq Khan, London's mayor, said on Twitter.
The prime minister is expected back in Brussels for further talks before the end of the week.