Speaking in the Isle of Man, where he was attending a meeting of the British-Irish Council, Lidington said he believed a "new dynamic" would emerge, once MPs saw the full text of the proposed agreement.
Downing Street has played down suggestions that a Brexit deal is imminent, after European Council president Donald Tusk appeared to indicate a breakthrough could come within the next week.
"That is why we're seeking on that is legally operative and one that gives us that guarantee that is necessary". "I think it's morelikely than not that well be able to include an agreement in the next few weeks before the end of the year".
Mr Varadkar said any deal would still have to be ratified by Westminster and the European Parliament and he said: "even when all of that is done then we begin the talks on the future relationship so there's no clean break here".
"The fact that Brexit is happening makes that hard to replicate, but our objective as an Irish Government is to do that to the extent that we can, in order to allow people to travel freely as they have done for so long now, but also to allow trade to function as it does now".
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"It's entirely appropriate that they should be communicating on an issue of real importance to all of us".
"But there are other political parties as well, who represent the majority of pepole in Northern Ireland, and there's also Northern Ireland business and Northern Ireland farmers, and people that live in Northern Ireland".
"We want to stick to the confidence and supply arrangement [with May's government] but you know there are two sides to that and the government has obligations under the confidence and supply arrangement as have we", Wilson said. "But Brexit has given rise to a hard situation and we need to resolve that". Nevertheless, London and Brussels still have not reached agreement on a number of issues, such as the Irish border and post-Brexit economic relations between the United Kingdom and the EU.
Democratic Unionist Party leader Arlene Foster told The Times that a letter she received from May this week about the Brexit Withdrawal Agreement "raises alarm bells for those who value the integrity of our precious Union".
This so-called "backstop to the backstop" would see Northern Ireland become wedded to the EU single market and customs union should London and Brussels fail to strike a permanent trade deal.
Sammy Wilson, the DUP's Brexit spokesperson, accused May of "total betrayal" and ditching the "promises she made" to the party, which has already threatened to withdraw its support for the embattled prime minister.