After days of political turmoil sparked by her botched gamble on a snap election, May's Conservative Party resumed talks with a small Northern Irish Protestant party on securing the support of its 10 members of parliament to pass legislation.
SNP leader Nicola Sturgeon also said she was concerned about the prospect of a DUP-Conservatives deal.
"We are continuing to have talks but today, as you will imagine, there has been a real focus on this awful tragedy in London", May said.
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".
The London fire, which claimed at least 17 lives, forced the cancellation of the City of London's Mansion House dinner, where May's finance minister, Philip Hammond, had the chance to revive calls for a more business-friendly exit from the EU.
In a letter to the Chancellor, Mr McDonnell said: "I am writing to ask whether the Government has been asked for this measure by the DUP, has agreed to it or is considering it".
Talks restarted on Monday in Belfast, under Britain's Northern Ireland Secretary James Brokenshire.
May needs the support of the DUP to govern, after a disastrous election result in which she lost her parliamentary majority. Though Foster campaigned for Brexit, voters in the province rejected it.
These concerns were echoed by former prime minister Sir John Major who has warned that the Government will compromise its stated impartiality if it enters a confidence and supply deal with the DUP at Westminster.
Cristiano Ronaldo in focus after Confederations Cup tie
Ronaldo's brilliant pass between three defenders let Ricardo Quaresma score the first for Portugal . We're very happy with this first game". (AP Photo/Martin Meissner).
"We remain fully committed to making the institutions work", she said.
"It's passing quicker than anyone believes". "We're ready to dance". Only time will tell.
Barnier dismissed the suggestion of postponing the negotiations and said such a delay would only prompt further instability.
France's Macron said the door was still open for Britain to remain in the European Union, though he added that it would be hard to walk back once negotiations start. They argue instead for a "soft" version, prioritising some form of continued access to the single market in order to minimise economic damage.
Sinn Fein, which won seven seats in the British parliament at last week's election but will maintain its policy of not taking them, said its leader in Northern Ireland, Michelle, O'Neill would repeat those concerns in London on Thursday.
Ms Foster dismissed suggestions her deal with the Tories threatened the peace process.
Her failure to win a majority has put May under pressure over her Brexit plans from inside and outside her party and has prompted complaints about her choice of partner due to the DUP's stance on social issues such as gay marriage.
"This new arrangement is very unsettling and people are concerned and anxious about what it may mean", Sinn Fein MP Michelle Glidernew told AFP.