Sinn Féin meanwhile is claiming the DUP turned its back on a draft deal that would have seen power-sharing restored.
Sources close to the negotiations told Reuters that some DUP members had issues with the proposed compromise and "robustly raised" their concerns earlier this week.
"We did have some conversations yesterday as well and I thought that was useful", she said before adding "but in fact, the visit yesterday became a bit of a distraction because we had still work to do and we'll continue to do that work today and in the coming days".
The row boils down to a row over the presentation of new laws to protect Irish language speakers.
Based on this opinion it seems clear that if Sinn Fein acquire a standalone Irish language act they will be legally in line with southern Ireland's constitution with predominance over other languages.
Declining to do so, they contend that revealing details could jeopardise any future talks. Its leader in Northern Ireland said direct rule was "not an option".
"This process has already taken too long".
Taoiseach Leo Varadkar said he "regretted" the breakdown in talks.
May said she'd urged the politicians "to make one final push for the sake of the people of Northern Ireland".
We need a budget for the Northern Ireland authorities and, obviously, at the moment I'm afraid we're not going to get it through a re-establishment of the devolved executive.
Foreign Secretary Boris Johnson failed to mention Northern Ireland or the post-Brexit border issue once in a speech responding to people's concerns about leaving the European Union hours earlier.
"Certainly there is no appetite to move towards direct rule (from London)..." Sinn Fein insists the Irish Government must have a key role in the region if Stormont does not return.
"They have now collapsed this process".
Theresa Villiers, the former Northern Ireland secretary, said the continued absence of a ruling executive in the province after more than a year of stalemate meant the UK Government would likely have to step in to keep services running.
"Those gaps were closed, that's why I don't understand (that) the commentary yesterday was as definitive as it was", Coveney said.
"We are ready to bring forward legislation to enable an executive to be formed", she said.
"There was a recognition that the Irish language needed to be legislated for in Northern Ireland but that should happen in the context of a basket of legislation that also dealt with issues like Ulster Scots and a broad recognition of culture and language diversity in Northern Ireland".