Arlene Foster and Nigel Dodds emerged from No 10 on Tuesday afternoon saying it had been a "very good meeting".
Responding to the story a DUP spokesman told the Belfast Telegraph there will be "all sorts of speculation and briefings". The UK prime minister insisted that he wanted an agreement to be struck with Brussels, admitting that he would be partly to blame if a deal could not be secured.
"We want the referendum result implemented".
"The Prime Minister has already ruled out a Northern Ireland-only backstop because it would be anti-democratic, unconstitutional and would mean our core industries would be subject to European Union rules without any means of changing them".
DUP leader Arlene Foster has criticised a Northern Ireland-only backstop as "anti-democratic and unconstitutional".
Varadkar also revealed that he had "not received" any legally-binding new proposals from the United Kingdom government for alternatives to the contentious Irish backstop, putting him at odds with Johnson's recent claims that progress was being made.
"History teaches us that any deal relating to Northern Ireland which can not command cross community support is doomed to failure", Mrs Foster said.
People born in Northern Ireland remain British citizens according to the law even if they identify as Irish, a court has heard..
"In the context of Northern Ireland, when one is interpreting any piece of legislation one must look at the Belfast Agreement and the provisions of it, and interpret the legislation in a way which is consistent with it", he added.
Government officials said Ms DeSouza is automatically British, having been born in Northern Ireland.
"The Prime Minister is 100% committed to getting a deal which works for the United Kingdom - and all of the United Kingdom - and abolishes the anti-democratic backstop", he added.