May is due to meet its leader Arlene Foster on Tuesday. "We're ready to dance".
Northern Ireland has been without a powersharing executive since March and without a first and deputy first minister since January, after Sinn Fein collapsed the administration amid faltering trust and relations with the DUP.
She has already met with the DUP to seek a deal to support a minority Conservative government.
"Discussions are going well with the government and we hope soon to be able to bring this work to a successful conclusion", Foster tweeted.
'Much of it is complete and utter nonsense, I have to say.
Sinn Fein's Stormont leader Michelle O'Neill has said she will raise the issue with Mrs May in face-to-face talks in London on Thursday.
In a hint at a softer approach, Mr Davis said on Monday "we will start down this process" by focusing on the divorce proceedings from the European Union before moving on to trade.
"The focus has to be on plan A, which is to get the institutions in place, that is our focus and we would like to think it will be the focus of the incoming taoiseach", he said. "I think it's a very damaging situation".
Brussels has insisted talks on the so-called divorce, taking in issues including the fee the United Kingdom will have to pay to sever its ties, must make sufficient progress before any discussion on a future trade agreement could begin.
He confirmed they had also discussed the possibility of a referendum on Irish unity.
The Conservative leader lost her parliamentary majority in last week's election and is now desperately seeking the backing of the 10 MPs from Northern Ireland's ultra- conservative Democratic Unionist Party (DUP).
"I will be making it very clear that any deal between the Tories and the DUP can not be allowed to undermine the Good Friday and subsequent agreements", she said.
He told the Press Association the Government could not act as a "neutral facilitator" in Northern Ireland, as the Good Friday Agreement envisages, if it was dependent on one of the Northern Irish parties for its majority in the House of Commons.
"The intent is to ensure that we have the stability of Government in the national interest". "What would happen then?"
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