Tusk's role in the delicate choreography of reaching an accord had been to wait until May strikes a deal on making "sufficient progress" on divorce terms with Juncker's Brexit negotiator Michel Barnier.
Juncker spoke first with Irish Prime Minister Leo Varadkar then with May on Thursday night in a bid to break a deadlock over the wording of a deal on future arrangements for the Irish border.
Prime Minister Theresa May travelled Friday morning with an offer: a financial settlement, an agreement on Europeans living in the United Kingdom and a solution for keeping open the border that divides the island of Ireland after the split.
"Specifically, more work is needed around the areas of cooperation where it would be necessary to have alignment of rules and standards", the DUP said, adding that more needed to be done to establish how alignment could be effected without staying in the EU's single market and customs union.
"Early morning meeting possible", the spokesman said on Twitter.
May has found it difficult to come up with a formula that satisfies both European Union member Ireland, which wants to avoid creation of a "hard" border, and Northern Ireland's DUP party, which says the British province must quit the European Union on the same terms as the rest of the UK.
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But more significantly - and perhaps crucially - the DUP's Westminster leader Nigel Dodds and the party's chief whip Sir Jeffrey Donaldson were seen emerging from 70 Whitehall, the offices of the PM's de-facto deputy, Damian Green.
European Union citizens within the United Kingdom "will be able to go on living as before".
Meanwhile, Confederation of British Industry (CBI) president Paul Drechsler has underlined businesses' "immediate" need for a transition deal, warning that 60% of firms with Brexit contingency plans will put them into effect by Easter if Mrs May does not get the green light for trade talks next week.
Simon Coveney said his administration would consider new proposals from London, but emphasised it would not agree to anything that fell short of the assurances it needs over the shape of the border post Brexit.
And the UK Foreign Secretary and leading Cabinet Brexiteer Boris Johnson insisted any deal must stick to the spirit of the Leave campaign.
Amid signals Britain could seek "regulatory alignment" with the European Union to ensure there does not need to be a hard border, Foreign Secretary Boris Johnson insisted any Brexit deal must mean "taking back control of our laws, of our borders and of our cash".