"Progress will not come from a deal between the DUP and Tories to prop up a government in Westminster with an austerity and Brexit agenda but through the full implementation of the agreements and an Executive that respects the rights and delivers for all in society".
The European Parliament's Brexit coordinator, Guy Verhofstadt, struck a harsher tone: Britain, he said, could change its mind, but it would be poorer.
Former Prime Minister John Major who was involved in talks with both sides of the Northern Irish conflict peace talks before handing over to Tony Blair has warned he has reservations about his party going into government with the DUP.
The Times newspaper said finance minister Philip Hammond would push May not to leave the customs union - an arrangement that guarantees tariff-free trade within the bloc but prohibits members from striking trade deals with third parties.
The report cited unnamed sources, and the finance ministry declined to comment.
Naomi Long of the non-sectarian Alliance Party said May did not reassure her of the government's neutrality in Northern Irish matters.
May's government has said its Brexit plans remain the same, and her Brexit minister David Davis will be pressing for close economic ties but a clear break with the bloc to be able to control immigration and restore sovereignty over British laws.
US jet shoots down Syrian warplane in Syria
Earlier, Syrian forces attacked coalition fighters in Ja'Din, wounding a number of fighters and driving them from the town. Military said it shot down the Syrian warplane, acting "in collective self-defense of Coalition partnered forces".
The PM has rejected claims the deal would undermine its duty to be impartial in Northern Ireland.
Speaking before she entered talks with the new Fine Geal Leader, Foster threw down the gauntlet to Sinn Fein and said her party were "ready to dance" in order to restore a power sharing executive up North.
While the DUP are deeply eurosceptic, they have balked at some of the practical implications of a so-called hard Brexit - including a potential loss of a "frictionless border" with the Republic of Ireland - and talks will touch on efforts to minimise the potential damage to Northern Ireland.
The DUP dislikes Jeremy Corbyn because of his support for the Northern Irish Sinn Fein Party, which is the DUP's main rival.
"We know each other and we understand each other", she said. "We will be asking the prime minister to be open with politicians and also with the public", SDLP leader Colum Eastwood said in a statement. Government bond prices suffered heavy losses on Tuesday after consumer price inflation jumped to 2.9 percent in May.
With the two-year clock on Brexit ticking since March, when a letter from May formally started proceedings, Barnier dismissed the suggestion of postponing the negotiations and said such a delay would only prompt further instability.
The original date for the Queen's speech was also supposed to be the day Brexit negotiations were due to begin in Brussels.