Trade minister Liam Fox described EU nationals in the UK as one of the government's "main bargaining chips" in upcoming negotiations, and May argued that the UK would be left "high and dry" in negotiations by guaranteeing the rights of European nationals without receiving similar assurances for UK nationals living in the EU.
"I think there is a unity of goal among people in the United Kingdom", May said following a meeting with French President Emmanuel Macron on Tuesday.
Media reports suggested an agreement could be delayed into next week, but the spokesman said: "I certainly have heard nothing on this side to indicate that".
May said on Tuesday that talks with the DUP had been productive - a view shared by DUP leader Arlene Foster - and that Brexit negotiations would begin as planned next week.
European Union officials see the start of Brexit talks on Monday as a sign Theresa May is accepting their format for negotiations but they expect no quick deals and are wary the prime minister may try to break with Brussels protocol.
Theresa May, the Prime Minister, wants Britain to leave both.
Britain will be the first member state ever to leave the bloc.
But a newly appointed junior Brexit minister, Steve Baker, told Reuters: "I don't foresee any change".
France's President Emmanuel Macron said on Tuesday that "the door remains open" for the United Kingdom to ditch plans for Brexit and stay in the EU.
Her spokesman said Britain would not change its stance on Brexit, though May told her lawmakers she would seek a broader consensus in the party on her approach.
"It will be a brand new door with a new Europe, a Europe without rebates, without complexity, with real power and with unity".
Irish nationalist party Sinn Fein said the prospect of a British agreement with the DUP was causing anxiety and fear.
"It is certainly politically desirable and provided for in the texts", she said. The UK leaving the European Union was "the most likely outcome", she added.
May gambled that a strong election win, as forecast by some pollsters, would boost her majority in the House of Commons in time for the Brexit talks, but instead, her party surrendered 13 seats in the lower chamber.
The EU's chief Brexit negotiator Michel Barnier also warned Tuesday that time was passing.
And as leaders welcomed the new tone in London and talk of a "softer Brexit" that may be less disruptive than May's clean break with the single market and customs union, officials from at least some governments saw compromise on the British bill.
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 minimize the potential damage to Northern Ireland.