The new proposal could revive talks that have been stalled for almost four months.
And in a remark that was welcomed by some Eurosceptics who want to scrap Mrs May's proposed trade plans, he said the European Union remained open to a "Canada plus plus plus" deal - a reference to the EU's free trade deal with Canada.
Diplomats have suggested the leaders could talk through the night and approve the outlines of an agreement while they are still in Brussels for the broader 18 October talks.
He said: "If we do negotiate a Canada +++. minus. asterisk. whatever.it's going to take the transition period to negotiate that agreement".
But for that to happen there would need to be a resolution to the major sticking point, the so-called "backstop" plan for Northern Ireland.
But he held out the offer a "Canada+++" trade deal, which was welcomed by many Brexit-supporting politicians as the basis for a final deal.More news: Republicans aim to confirm Brett Kavanaugh this weekend
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But at a press briefing in Brussels on Thursday after a meeting with Irish Taoiseach Leo Varadkar, Tusk - the former prime minister of Poland - said: "In respecting our partners, we expect the same in return".
European Union diplomats and officials have also said they have not seen any formal new written text from London.
The prime minister says she is opposed to such a deal precisely because it would require a customs border in the Irish Sea, and would lead to the break-up of the United Kingdom as we know it because Northern Ireland would have to be treated differently to the rest of the UK.
Officials in Brussels expect a fresh proposal from Britain on the Irish Border after an earlier London paper was rejected by European Union leaders.
The EU's proposed backstop solution would see Northern Ireland stick to those rules of the customs union and single market that are required for cross-border co-operation to continue. The UK government has acknowledged that could leave planes grounded and trucks backed up at British ports.
Juncker added that he hoped the European Council later this month would make "enough progress" that "we can see it through in November".
Senior Democratic Unionist, Sammy Wilson, one of the DUP's 10 MPs at Westminster who are holding up the minority Government, went on Twitter. They are implacably opposed to any deal that separates Northern Ireland from Britain - and some kind of checks between the two may be needed in order to get a deal that avoids a new border emerging on the island of Ireland. Much further-reaching on trade, on internal security and on foreign policy cooperation.