However, Sinn Fein's leader in Northern Ireland, Michelle O'Neill, said the two sides had reached an "accommodation" on this and other contentious questions but said the DUP failed to follow it through.
SINN FEIN will meet with leaders of both the Irish and United Kingdom governments today to discuss the failed bid to restore power-sharing in the North.
The DUP has rejected the claims, insisting no final deal was on the table.
"We advised the DUP leadership that the deal should be closed before those opposed to it could unpick what we had achieved", she said.
The political crisis threatens power-sharing, the key achievement of the 1998 peace accord that ended decades of violence and bloodshed in Northern Ireland.
As it stands Mrs Bradley is under a legal obligation to call another snap election in Northern Ireland but few observers see the merit in such an option, given it would likely return the same political make-up and not resolve any of the outstanding issues.
Foster said Wednesday the DUP was willing to "to reach an accommodation on language and cultural issues".
Image Mrs May with Bombardier president Michael Ryan during her visit
Many unionists see the promotion of Irish as an attempt to undermine Northern Ireland's British identity, and as a step toward Sinn Fein's ultimate goal of joining it with the Republic of Ireland.
Same-sex marriage was also an obstacle to the two parties - it's illegal in Northern Ireland, but legal in the rest of the United Kingdom and in Ireland.
Hopes of a deal were raised on Monday when British Prime Minister Theresa May and her Irish counterpart, Leo Varadkar, flew in for talks on Monday, but Foster later said their presence was a "distraction".
Several UK-government-set deadlines to restore the Northern Ireland administration had passed without success, raising the spectre that the British government might impose direct rule from London on Northern Ireland.
The vacuum has limited Belfast's say in Brexit negotiations that will decide the future of the border between the North and EU-member Ireland.
May said she'd urged the politicians "to make one final push for the sake of the people of Northern Ireland".More news: Paul Dummett: Newcastle United left-back signs new deal until 2022
More news: Meek Mill's Lawyers File Documents To Nullify His Sentence
More news: Facebook launches Messenger Kids, its child-friendly texting app on Android