The move is part of the DUP protest against the Northern Ireland Protocol.
The protocol is the part of the Brexit deal that applies to the North of Ireland.
The move would mean Deputy First Minister Michelle O’Neill, of Sinn Féin, also loses her position.
Due to the North’s power-sharing arrangements, the roles of first and deputy first ministers are a joint office shared between the two biggest parties at Stormont.
Neither leader can stay in power if the other person resigns.
There has been no official statement from the DUP.
But sources told the BBC’s Nolan Show an announcement is expected.
It is not clear whether Mr Givan will stand down immediately or announce his resignation for a future date.
Speaking to the Nolan Show, the DUP Minister Edwin Poots refused to confirm that the resignation would be made public on Thursday, simply saying that people would “just have to wait”.
If the move happens it comes just hours after Mr Poots announced that he was stopping Irish Sea border checks as part of the Northern Ireland Protocol.
The DUP are the biggest party in Stormont and so they are entitled to hold the first minister’s post.
But it was being predicted that if the Assembly elections were held this May, Sinn Fein would have become the biggest party with Michelle O’Neill to take over the mantle as First Minister.
Mr Givan was appointed first minister just over eight months ago in June 2021, during a turbulent period for the DUP, in which the party had three different leaders with in a matter of weeks.
The party’s current leader, Sir Jeffrey Donaldson, has been threatening to withdrawn DUP ministers from Stormont for months in protest at the NI Protocol.
There has been speculation that Donaldson would collapse the Assembly on or before February 21.
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