The Republic of Ireland's foreign affairs minister, Simon Coveney, said the "last thing" Northern Ireland needs is for its largest party to be divided.
So far, Givan is now is expected to be sworn into office as the new first minister, with Sinn Fein's Michelle O'Neill as deputy first minister.
The meetings came amid a stand-off between Sinn Fein and the DUP over the appointment of a new First and deputy First Minister, and the introduction of Irish language legislation provided for in the New Decade, New Approach deal.
It is unlikely a new DUP leader would keep him in position and he could well choose to quit of his own accord before he is replaced.
The latter was said to gain criticism from his party for allowing major concessions to DUP's main opponents, the Northern Ireland National Republican party, Sinn Fein, who seek more independence from London, so that nationalists support Givan's candidacy.
He narrowly defeated Sir Jeffrey Donaldson in the leadership contest, and there has been speculation that the Lagan Valley MP is now the front-runner to succeed Mr Poots.
An electoral college made up of DUP MLAs and MPs will decide who will lead the party next.
"We'll have to wait and see what happens in relation to Paul Givan as First Minister".
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Political turbulence comes amid growing dissatisfaction among Unionists in Northern Ireland following Brexit as, under the terms of the withdrawal agreement, a customs border has been established between the nation and the rest of the UK.
Mr Poots will stay in his post until a successor is elected. We will work with the new leadership - that is a matter for the DUP - and whoever emerges as their leader. "It's been a very hard time for the outgoing leader Edwin Poots but I think it's important that we maintain relationships".