Edwin Poots will not guarantee Irish language law this term
DUP leader Edwin Poots has said he intends to support Irish language legislation, but not necessarily before the next assembly election.
Sinn Féin has suggested it may not support a new DUP first minister unless such legislation is approved by ministers before 10 July.
Conor Murphy said his party had not been assured that Irish language commitments would be honoured.
Arlene Foster resigned as first minister at 13:00 BST on Monday.
Mrs Foster made a statement to the Northern Ireland Assembly before her resignation, urging political parties to stick to a language and cultural deal they agreed to in January 2020.
She said the deal was "the only model for success".
Her departure comes six weeks after she was overthrown as the leader of the DUP.
Michelle O'Neill lost her position as deputy first minister when Mrs Foster resigned because the roles are a joint office.
There is now a seven-day period for the approval of a new first minister and deputy first minister by Sinn Féin and the Democratic Unionist Party (DUP).
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Mr Poots has said he will nominate his colleague Paul Givan to become the new first minister.
Mr Poots added that conditions could not be attached to nominating a first and deputy first minister.
"Setting pre-conditions is not appropriate, it's not respecting someone's mandate, and we cannot be in a circumstance where we have pre-conditions set for the selection of our first minister. And I'm not setting pre-conditions to the selection of Sinn Fein's deputy first minister," he said.
Sinn Féin must nominate a candidate for deputy first minister or a first minister is unable to take office.
Mr Murphy said Sinn Féin had "run out of road" with the DUP regarding commitments to implement Irish language legislation.
He told a press conference at Stormont that the previous DUP leadership had undertaken to bring the legislation forward in December, January and February but no progress had been made.
He said Sinn Féin had told the British and Irish governments recent talks with the new DUP leader had gone as far as they could.
"We've tried our best in good faith.. but the DUP are being disingenuous," added Mr Murphy.
He said Sinn Féin wanted to see a resolution but added that "broken commitments undermine seriously the ability to do power-sharing".
He did not clarify if Sinn Féin would refuse to nominate a deputy first minister unless Irish language legislation is introduced to the assembly before it breaks for summer recess on 10 July.
Earlier on Monday, Mr Poots said he remained committed to the New Decade, New Approach agreement.
The deal included an Office of Identity and Cultural Expression, an Irish language commissioner and a commissioner to enhance and develop the language and culture of the Ulster-Scots/Ulster-British tradition.
However, he said "time will tell" whether it would happen in the current assembly term, which is due to expire in April 2022.
If a new first minister and deputy first minister are not elected by Thursday then the planned meeting of the executive that day cannot go ahead.
This would mean the latest planned relaxation of Covid restrictions - including the return of live music from 21 June - could not be signed off by ministers.
Individual ministers can remain in post to take day-to-day decisions for their departments, but they cannot bring new legislation to the assembly and any significant or controversial matters requiring executive approval would effectively be on hold.
It is the outcome nobody wants but avoiding it will be a challenge
Sinn Féin and the DUP are heading for the trenches to battle it out over Irish language legislation again
History tells us such battles can easily end with a collapse of the institutions, putting us on the path for an election
Sinn Féin wants Irish language legislation approved by the executive before the summer recess on 10 July.
When asked if that will happen, the new DUP leader said "time will tell" and when pushed said "not necessarily".
Without a definite "no", it allows some political wriggle room
But Edwin Poots also knows if Sinn Féin refuse to nominate a deputy first minister and trigger a crisis leading to a collapse, there will be a price to pay for the party
It will now become a game of political poker and the stakes couldn't be higher
Mrs Foster has confirmed she will leave the DUP once her time as first minister has ended.
She served as first minister from January 2016 to January 2017 - when the assembly collapsed - and again from January 2020.
Her departure has prompted a series of resignations within the DUP, with several councillors and party members expressing their anger about how she was ousted.
On Sunday, a decorated Army veteran turned DUP councillor was the latest to announce he was resigning from the party.
Ryan McCready, 35, who sits on Derry City and Strabane District Council, said the new leadership's outlook was "no longer compatible" with his own beliefs, citing the party's treatment of Mrs Foster as a reason for his resignation.
"I understand that my decision may upset and offend some of you, but I must put principle above all else," Mr McCready said in a statement on Monday.
Mrs Foster stood down as DUP leader after an internal party revolt, with at least 22 assembly members (MLAs) and four MPs signing a letter of no confidence in her.
Mr Poots has consistently played down talk that the party is split.
Last week he announced his ministerial team, but was criticised by two departing DUP ministers who said his selections did little to heal the division within the party.
Speaking last week, Mrs Foster said she was sad about the manner of her departure but was "looking forward" to new possibilities.