Barnier held "talks about talks" with May's Brexit advisor Olly Robbins and British EU Ambassador Tim Barrow in Brussels on Monday but they failed to agree on a date for the negotiations to begin, an EU official said.
It says he would seek support from the prime minister's old friend, ally and newly appointed deputy, Damian Green.
Conservatives and the DUP are committed to strengthening British unity, delivering Brexit, combating terrorism and delivering prosperity but "at the moment there isn t a deal", the source said.
But the UK Government has said it wants the talks to take place in parallel during the Brexit process.
Britain's prime minister has opened talks with a Northern Ireland-based party Tuesday to see if they can create an alliance to push through the Conservative Party's agenda after a disastrous snap election left her short of a majority in Parliament.
The state opening will be followed by days of parliamentary debate and a confidence vote that will be the first major challenge for the new government.
May's office said Northern Ireland's five main parties would take part, but a spokesman for the DUP did not immediately respond to a request to confirm their participation.
It has always been speculated that such an agreement would leave the government in breach of its duties under the Nothern Irish power sharing arrangement, signed on Good Friday in 1998.
"The risk is that Northern Ireland continues to fail to find solutions, and potentially the peace process unwinds", Simon Usherwood, senior politics lecturer at Surrey University, told AFP.
Several Conservative and other leaders have cautioned against the coalition since it would compromise London's neutral position between the contending groups in Northern Ireland - the Republicans and Unionists - and adversely hit the Good Friday Agreement.
Tunisia backs Turkey's role as mediator in Gulf-Qatar crisis
Qatari officials did not immediately respond to questions Thursday, though a State Department official connected the two sales. It has forced us to assert that no other nation can dictate to the United States how we will address such disputes.
An election in March saw the Protestant, pro-British DUP finish narrowly ahead of Catholic socialists Sinn Fein.
The talks come in the context of a British government-imposed deadline of June 29th to reach a deal on power-sharing or see the re-introduction of direct rule from Westminister.
Brexit is a complicating factor in the mix.
Asked about Mr Verhofstadt's comments, European Commission spokesman Alexander Winterstein played down suggestions of a change in the UK's status, telling reporters: "The UK is a member of the European Union and will remain so until it leaves, so that is where we are".
In the run-up to the start of negotiations with her European Union partners to agree the terms of the UK's departure, May has embraced a confrontational stance in obeisance to the anti-Europe wing of her own party and an anti-Europe right-wing press for whom putting Johnny Foreigner in his rightful place is long overdue.
The prime minister made the commitment during separate meetings at 10 Downing Street with the DUP, Sinn Féin, the SDLP, the Ulster Unionists and Alliance on Thursday.
Speaking in London Thursday after a meeting with May, prominent Northern Irish politician and Sinn Fein leader Gerry Adams said he told May she was "in breach" of the Good Friday Agreement - the peace treaty which ended 30 years of deadly sectarian violence known as "The Troubles".
The Commission said the high-priority issue of the shared border between Northern Ireland, which is part of the United Kingdom, and Ireland, an European Union member, will also be discussed.
"The prime minister has totally ignored the will of those people in terms of Brexit".