The PM has secured "private concessions" from Brussels that the whole of the United Kingdom will be allowed to remain in a customs union with the EU after Brexit occurs in March, according to The Sunday Times' political editor Tim Shipman.
Mr Donaldson said a no deal would have "serious consequences" for the Republic of Ireland's economy and that the United Kingdom "won't have to pay a penny more" to the EU. She said that, while the United Kingdom should aim to secure a withdrawal agreement as soon as possible, this would not be done at any cost.
The Prime Minister said any agreement will be dependent on an "acceptable" framework for future relations in areas like trade and security, expected to be covered in a separate political declaration.
The crunch meeting comes amid mounting pressure to finalise the withdrawal agreement in order to announce the leaving terms at a special summit in November.More news: 'The hottest thing': Trump launches final day blitz on Democrats
"We're willing to consider improvements to the backstop but we need to reach an agreement for this backstop and this backstop must be a genuine backstop".
Following the Cabinet meeting, Mrs May's spokesman said: "The Prime Minister said she was confident of reaching a deal".
In response to tweet from Irish foreign minister Simon Coveney on Monday, the DUP's chief whip in the House of Commons, Jeffrey Donaldson, cautioned that it "Looks like we're heading for no-deal".More news: High court declines to extend halt to climate change lawsuit
He argued that Britain would effectively become a non-voting member of the European Union, having to accept laws made in Brussels with no power to influence them.
The optimism was fuelled by growing hopes of a Brexit deal breakthrough after a cabinet meeting.
"We are coming to a stage where the markets believe that a deal may be on the cards and sentiment is not that negative towards sterling as it was a few weeks ago", said Manuel Oliveri, an FX strategist at Credit Agricole in London.
Donaldson said that he could not understand why the Irish government "seems so intent on this course".
A no-deal outcome, he said, "will have serious consequences for economy of Irish Republic".More news: Man fighting for life after shark attack in Whitsundays
Cautious optimism that a deal may be in the offing has also been dampened by uncertainty over whether it would pass the British parliament, deeply split between eurosceptic and pro-EU lawmakers, even within May's Conservative Party. In addition, United Kingdom won't have to pay a penny more to European Union, which means big increase for Dublin.