Meanwhile, Foreign Office Minister Sir Alan Duncan said Tory MPs seeking to oust the Prime Minister over Brexit represented a fringe element.
Senior figures in Brussels said that there was now "goodwill on both sides" and a realistic plan to solve the backstop question was being worked on.Sabine Weyand, the EU's deputy Brexit negotiator, told a meeting of European Union ambassadors: "Things have started moving".
"We want there to be decisive progress at the October summit, so that allows us to seal a deal in November", he said.
Mrs May will hope her updated offer to the European Union will give them more time to resolve the Irish backstop, which would come into effect should not deal be reached.
But last month, the EU's chief negotiator Michel Barnier said that if both sides were "realistic" there could be an agreement by November, when a special one-off summit has been arranged.
"We have agreed in principle a transition period".
Brexiteers have jumped on the EU's offer of a "Canada +++" free trade deal for the United Kingdom once it quits the bloc, piling more pressure on Theresa May to ditch her compromise Chequers deal.
Following a meeting with Irish Taoiseach Leo Varadkar, Mr Tusk took aim at the UK Government saying that overheated rhetoric "will achieve nothing except wasting more time" during Brexit negotiations.
Speaking at the Conservative party conference in Birmingham on Tuesday, Ms Foster stressed her party's strong objection to an Irish Sea Border: "The red line is blood red, it is very red".
"So all of those things are Ireland's objectives and we'll continue to press them as we go forward into the final phase of these talks".
Dominic Raab was quizzed over the DUP and the Irish border.
"We want to have a deal".
Appealing to the public for help with funding the legal challenge, Mr Maugham, director of the Good Law Project, said: "This is a case vital in our national interest".
The summit on October 18 to 19 has been billed as the "moment of truth" when it will become clear whether it is possible for the two sides to do a deal.
"What I do know is that we need a backstop, a protocol on Ireland and Northern Ireland as part of a withdrawal agreement", he said, adding "I think we are entering a critical and decisive stage of these negotiations and there is a good opportunity to clinch a deal over the next couple of weeks".