It comes just days after Mr Givan was officially installed in the post.
Mr Givan was informed of the news by DUP officers after Edwin Poots resigned as party leader.
He stepped down on Thursday after an internal party revolt over his decision to agree a deal with Sinn Féin and nominate Mr Givan as first minister.
The story was first reported by the Belfast Telegraph.
Mr Givan is Northern Ireland’s youngest first minister and looks set to be the shortest serving leader to hold the post.
Sir Jeffrey Donaldson, who is favourite to be the next DUP leader, has made it clear he plans to lead the party from Stormont as first minister.
The Lagan Valley MP ran against Mr Poots for the leadership and was narrowly defeated in the wake of Arlene Foster’s ousting.
When Mr Givan resigns, it once more will see Michelle O’Neill removed as deputy first minister and the DUP and Sinn Féin will then have seven days to fill the roles.
Mr Poots, who took over as leader in May amid internal tensions about the direction of his leadership and the manner of Mrs Foster’s departure, announced his resignation after a meeting of DUP party officers on Thursday night.
A motion of no confidence was mentioned at the meeting, but not tabled.
Speaking to RTÉ, Minister for Foreign Affairs Simon Coveney said that the DUP has “had a terrible number of weeks”.
“I think they will now look to putting a new leader in place quickly and to provide stability behind that person and we need to give them the space and time to do that,” he said.
“I also think other parties in the executive will be responsible in terms of trying to get that new first minister in place, if there is a new first minister, or if Paul Givan remains in office.”
This latest issue was sparked after a late-night deal was done between Sinn Féin, the DUP and Secretary of State Brandon Lewis ending a stalemate over Irish language laws to allow for the nomination of a new first minister.The position had become vacant after Mrs Foster formally resigned on Monday, thereby also triggering the resignation of Sinn Féin Deputy First Minister Michelle O’Neill, as it is a joint office.
Mr Poots had named Mr Givan as the DUP’s pick to replace Mrs Foster, while Sinn Féin said Ms O’Neill would resume her post as deputy first minister.
But Sinn Féin had wanted commitments from the DUP over a timetable for implementing Irish language legislation, as set out in the New Decade, New Approach (NDNA) deal that restored power sharing in January 2020.
The party said this was not forthcoming, and it called on Mr Lewis to bring in Irish language legislation via Westminster.
Mr Poots had said he was committed to implementing all of the NDNA commitments, including those around Irish language, but not necessarily before the end of the current assembly mandate in May 2022.
A deal was then struck between the two parties and the secretary of state in the early hours of Thursday morning that would see Westminster legislate for Irish language commitments in October, if Stormont fails to do so beforehand.
DUP MPs and peers wrote to Mr Poots expressing concern about the deal, ahead of a DUP assembly group meeting which it is understood saw the vast majority of DUP MLAs – 24 to four – oppose Mr Poots nominating Mr Givan as first minister as a result of the deal.
However, Mr Poots and Mr Givan left the meeting early, ahead of a special sitting of the assembly which then confirmed Mr Givan and Ms O’Neill as first and deputy first ministers.
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