The DUP also urged the Government to give "greater focus".
On the first day of talks, on Monday, it was decided that the highest-ranking officials from both sides - Oliver Robbins and Sabine Weyand - would oversee the issue - a signal of how seriously the United Kingdom and the European Union take the matter.
Theresa May has promised to work with "humility and resolve" as she prepares to set out her legislative programme for the next two years in a Queen's Speech expected to be dominated by Brexit.
The source declined to discuss BBC reports that a deal could be reached as early as Thursday.
May, who called a snap election expecting to fatten her party's majority, is promising her minority government will be one "that consults and listens".
A DUP source claimed an agreement was "certainly not imminent".
"The fact is that over 80 per cent of the electorate backed the two major parties, both of whom campaigned on manifestos that said we should honour the democratic decision of the British people", May said.
Deadly London tower blaze began in a Hotpoint fridge freezer
Eleven buildings around Britain have tested positive for the combustible cladding found in the Grenfell Tower . Police said the official number of victims has risen to nine but that the presumed death toll remains 79.
After initial signs of progress, the Conservatives were forced to row back on a premature announcement that agreement had been reached, and talks have now dragged on for 11 days without reaching a conclusion.
Arlene Foster's Democratic Unionist Party (DUP) is reportedly demanding billions be pumped into Northern Ireland in exchange for Tory political support. And although the unionists are supportive of Brexit, they want guarantees of freedom of movement and an open border with the Republic.
But Conservative Party sources told the BBC that a last minute deal is still possible.
"We are putting our shoulder to the wheel to try to get a good deal in London and also get a good deal here in Belfast as well". "We will be able to provide you with a solution that will help you make the right choice".
Of 27 Bills and draft bills unveiled in her first Queen's Speech, eight are devoted to the complex process of withdrawal from the EU, including a Repeal Bill to overturn the 1972 Act which took Britain into the European Economic Community and separate Bills on customs, trade, immigration, fisheries, agriculture, nuclear safeguards and the global sanctions regime.
"Theresa May has no mandate for the direction she is taking the country".
The Times branded her administration the "stumbling husk of a zombie government" and said May was now "so weak that she can not arbitrate between squabbling cabinet ministers", who are increasingly publicly divided over Brexit.
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