Theresa May is under increasing pressure from all sides to break the impasse in the Brexit talks.
The DUP says there is “work to be done” if it is to agree to plans for the future of the border with Ireland.
Irish PM Leo Varadkar said he was willing to consider new proposals and suggested the UK might put something forward later.
Ambassadors of the 27 EU member states are understood to be “waiting for something from London”.
On Monday, Northern Ireland’s Democratic Unionist Party – whose support the PM needs to win key votes at Westminster – objected to draft plans drawn up by the UK and the EU.
The DUP said the proposals, which aimed to avoid a “hard border” by aligning regulations on both sides of the border between Northern Ireland and the Republic of Ireland, were not acceptable.
The party has said it will not accept any agreement in which Northern Ireland is treated differently from the rest of the UK.
The DUP’s deputy leader Nigel Dodds said the Irish government, which has said it wants firm guarantees that a hard border can be avoided, was playing a “dangerous game” with its own economy.
At a press conference with his Dutch counterpart on Wednesday, Mr Varadkar said: “Having consulted with people in London, she (Theresa May) wants to come back to us with some text tonight or tomorrow… I expressed my willingness to consider that.”
The BBC’s Adam Fleming said that following an update from chief EU negotiator Michel Barnier on Wednesday, EU member states agreed there must be clarity within 48 hours for them to have enough time to consult with their capitals about draft guidelines for phase two of the talks.
At a summit next week, European leaders will decide whether enough progress has been made in the negotiations on Ireland, the UK’s “divorce bill” and citizens’ rights so far to open trade talks.
Amid the race to find an agreement suiting all sides, Brexit Secretary David Davis told MPs on Wednesday that the government had not carried out any impact assessments of leaving the EU on the UK economy.
Mr Davis said the usefulness of such assessments would be “near zero” because of the scale of change Brexit is likely to cause.
He had previously said the government had done 57 studies on 85% of the UK economy about the impact of Brexit.
Meanwhile, on Wednesday, 19 Tory MPs who back a “soft Brexit” wrote to Mrs May saying it was “highly irresponsible” for anyone to dictate terms which may scupper a deal.
It followed some Tories backing the DUP’s decision to oppose a draft deal on the future of the Irish border.
In their letter, the 19 MPs – who largely backed Remain in the 2016 referendum – say they support the PM’s handling of the negotiations, in particular the “political and practical difficulties” relating to the Irish border.
But they hit out at what they say are attempts by some in their party to paint a no-deal scenario in which the UK failed to agree a trade agreement as “some status quo which the UK simply opts to adopt”.
The MPs included former cabinet ministers Stephen Crabb, Dominic Grieve, Anna Soubry and Nicky Morgan.