Tusk said: “The key to the UK’s future lies in some ways in Dublin, at least as long as Brexit negotiations continue.”
Varadkar warned: “I am also prepared to stand firm if the offer falls short.”
He said the next couple of days will be crucial, the Press Association reported.
Theresa May is hoping EU leaders will give the go-ahead for the second phase of the negotiations, including talks on a free trade deal, to begin at the European Council summit on December 14-15.
She is due to travel to Brussels on Monday for talks with European Commission president Jean-Claude Juncker in the hope of securing a declaration that “sufficient progress” has been made on divorce issues like the financial settlement and the Irish border.
The leaders of the remaining 27 EU states, including Irish premier Varadkar, have a veto on triggering the second phase of talks, meaning May must be sure of support from Dublin for progress to be made.
Tusk confirmed the Irish would be consulted on whether the UK’s offer was sufficient.
He added: “If the UK offer is unacceptable for Ireland it will also be unacceptable for the EU.
“I realise that for some British politicians this may be hard to understand but such is the logic behind the fact that Ireland is an EU member while the UK is leaving.”
The free-flowing and invisible border between Northern Ireland and the Republic since the 1998 peace deal was enabled through the UK and Ireland’s joint membership of the EU and its single market.
When the UK departs it raises the prospect of some form of customs barrier or else continued harmonisation on issues like duties on goods.
Ireland’s proposal for no checkpoints could mean Northern Ireland sticking to the EU rulebook while the rest of the UK diverged, a red line for the Democratic Unionists who are propping up the UK Government.
Tusk met the Taoiseach in Dublin on Friday afternoon.
Varadkar said good progress had been made during EU-UK negotiations on finance and EU citizens rights as well as on people moving between Ireland and the UK.
He said advances had been made towards a common understanding on protecting the 1998 Good Friday Agreement which ended violence in Northern Ireland.
But he warned that the British Government must offer credible, concrete and workable solutions guaranteeing no hard border.
He said: “I am also prepared to stand firm with our partners if needs be, if the UK offer falls short on any of those key issues, including the Irish ones.”
The UK is pressing for a frictionless frontier on its only land border with an EU state.
It wants to use technology and trading arrangements to ensure the free flow of goods and no return to the heavily militarised border of the conflict.
The Taoiseach said the UK was Ireland’s friend and neigbour and they wanted to work together.
He added: “I want to make progress but I also need to make clear: the EU 27 cannot declare sufficient progress without firm and acceptable commitments on the border and that is a position shared across the political spectrum in Ireland.”
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