Adams says Ireland's incoming government must act to protect the peace process.
Following March's snap Assembly election, the DUP no longer has the electoral strength to deploy a petition of concern in its own right, though it could still potentially combine with other socially conservative MLAs to do so.
Davis also sought to reassure the public that any deal between the Tories and the Democratic Unionist Party would not change abortion or gay rights laws.
The dispute has prompted renewed calls for a chair from outside the United Kingdom and Ireland to be appointed.
"Northern Ireland's political leaders now have this chance to take control and restore effective power sharing government under the current assembly mandate".
European Commission spokesman Alexander Winterstein said "we are quite confident" that technical talks "can start soon, maybe even this week".
Meanwhile British Prime Minister Theresa May is already in talks with the northern Ireland politician to see if they can formulate a government.
There are fears the central role being offered to the DUP could call into question the "rigorous impartiality" expected from Mrs May's administration under the Good Friday Agreement in the event of political disputes in the North.
Trump says he's willing to testify under oath, contradicts Comey
The former Federal Bureau of Investigation director detailed "disturbing" interactions with Trump that led to his firing on May 9. In his testimony, Comey said Trump sought a pledge of loyalty, and told Comey he hoped he would drop the Flynn investigation.
The EU has said that Brexit talks need to make sufficient progress before trade deals can be discussed, though Britain had argued the discussions should take place simultaneously.
A small political party entirely based in Northern Ireland that has been out of the spotlight for years could emerge as a decisive force in allowing the Conservative Party to form a majority in the House of Commons after Theresa May's disastrous campaign result in the 2017 general election.
Northern Ireland is the only part of the British Isles where same-sex marriage remains outlawed. She said Sinn Féin is keen to re-enter the Stormont executive.
Enda Kenny has warned the British Prime Minister that nothing should happen to put the Good Friday Agreement at risk. All those involved must engage in the full knowledge that the deadline of 29 June is final and immovable.
He made clear the reintroduction of direct rule from Westminster is on the cards if an agreement does not materialise by that date.
The Prime Minister has now appointed the Rt Hon James Brokenshire MP as Secretary of State for Northern Ireland.
The institutions collapsed after the late Martin McGuinness quit as deputy first minister in protest at the DUP's handling of the ill-fated renewable heat incentive (RHI) - an eco-scheme that left Stormont facing a £490 million overspend.
The party is now led by Arlene Foster, a lawyer whose father was shot in an IRA gun attack when she was a child.
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