Sinn Féin leader Mary Lou McDonald has said a draft agreement was in place last week to restore Northern Ireland's power-sharing executive.
The talks ended on Wednesday after the DUP declared there was "no current prospect" of a deal.
Ms McDonald said the agreement included an Irish language Act and the DUP was warned to "close the deal before those opposed to it could unpick" it.
However, DUP leader Arlene Foster said that no draft agreement was in place.
Mrs Foster told Sky News that Sinn Féin "certainly didn't have an offer of an Irish language act".
"We didn't reach an agreement," she said.
"I regret that we didn't reach an agreement - they were insisting that they have this stand-alone Irish language act and that is not something I could sign up to - I have always been very clear about that."
The Prime Minister spoke to the leaders of both Sinn Féin and the DUP on Thursday evening.
A spokesperson for the Prime Minister said Mrs May "expressed her disappointment" that an agreement had not been reached.
"The Prime Minister was clear that she still believes the basis for an agreement exists, and urged them both to reflect on the recent phase of talks in order to find the best way forward".
Sinn Féin and the DUP had been in negotiations in a bid to end the 13-month stalemate at Stormont.
Mrs Foster dismissed claims that she had lost control of her party as "nonsense" and said that she had kept her party officers briefed throughout the negotiations.
"We have had a number of meetings so I could bring them up to date as to where we are at, but I have never felt that I am in a position that I can make a recommendation and so no recommendation has been made to my officer board," she said.
Mrs Foster told RTÉ that a "completely different set of circumstances" existed in Northern Ireland than in Scotland or Wales, because "in those places language has not been used as a political demand and that needs to be recognised.
"They need to recognise that the majority of people here do not want an Irish language act a la Wales," she said.
'Recognition of Irish'
Ms McDonald claimed a draft package in place last week included an Irish language act, an Ulster Scots act, and respecting language and diversity act.
"The Irish language act included provision for official recognition of Irish, the creation of an Irish language commissioner, the repeal of the ban on Irish in the courts was also to be legislated for," the Sinn Féin leader said.
"It did not involve at any stage making Irish compulsory or applying quotas to public services. This was not a consideration," she added.
Ms McDonald said there had been "no meetings of mind on marriage equality", but it was agreed that it would be considered in the assembly in the form of a private member's bill.
The Sinn Féin leader said the DUP must now explain their position.
"The DUP leader brought this phase of negotiations to a close and said 'there is no current prospect of these discussions leading to an Executive being formed'," she said.
"It is up to Arlene Foster to explain this given that the DUP and Sinn Féin leaderships had achieved an accommodation across the issues involved."