The two sides are trying to push the divorce deal as well as an agreement on post-Brexit relations above the line in time for two leaders' summits scheduled for October 17-18 and November 17-18.
Jacob Rees-Mogg, the leader of the pro-Brexit Tory European Research Group, said it represented "a good solution for everyone", while Sir Bernard Jenkin also welcomed Tusk's offer, but said there must be a satisfactory solution to the Irish border issue.
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. But our will is unbroken to reach agreement with the British government.
A senior Whitehall source said: "This is gamesmanship by the EU". The EU is serious about getting the best possible deal.
The deal appears to be dependent on Mrs May accepting different customs arrangements in Northern Ireland and mainland Britain, which she has been clear she will not accept.
"To sum it up: Anyone who thought that an end of the Brexit drama is in sight will have to think again", said analyst Thu Lan Nguyen.
"That's how it was in Salzburg, and that's also how we will work in the coming days".
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Senior diplomats said there was a mood of "cautious optimism" on Friday evening after members of Brexit negotiator Michel Barnier's team briefed them on "signals" that the United Kingdom was ready to come forward with a new Irish border plan.
Mr Varadkar said that he did not know what a "Canada plus plus plus" deal meant, but added that it could take longer than the length of the post-Brexit transition period to agree a final EU-UK trade deal.
He and Varadkar, he said, had "focused on practical and realistic ways of minimising the damage caused by Brexit, on both sides of the channel".
Tory Brexiteers - who say Chequers would keep Britain too closely tied to the European Union after it has left the bloc - warmly welcomed Mr Tusk's comments saying they showed another deal was possible.
The add-ons will include counter-terrorism cooperation, foreign policy and defence coordination, and other ways in which Britain and the European Union will work together, such as on recognition of each other's social security and qualifications systems, agreements on transport connectivity and some reciprocal arrangements to allow people to work in each other's territories.
A majority of the Cabinet reportedly now supports moving towards a Canada-style trade deal over Mrs May's plan to keep the country tied to European Union rules on goods, therefore hindering an independent trade policy.