Solskjaer, in charge of Norwegian club Molde, was instantly installed as favourite to take over at Cardiff City Stadium when Malky Mackay was sacked on Friday following a series of disputes with owner Vincent Tan.
Chairman Mehmet Dalman has stated he has one candidate in mind and refused to deny he had held discussions with Solskjaer, saying: " I might have and I might not have".
There have been suggestions Solksjaer has turned down Dalman's overtures, but Press Association Sport understands Solskjaer has yet to reject the chance to take over in the Welsh capital, and that Dalman hopes the promise of funds to spend in January and assurances regarding the working structure at the club will convince the 40-year-old to make the move to the Premier League.
With Dalman's recruitment hunt ongoing, Mackay's assistant David Kerslake and first-team coach Joe McBride remain in charge of the Bluebirds for the daunting New Year's Day trip to leaders Arsenal.
Kerslake insists he knows nothing about the ongoing hunt for a new manager, but has called on the club's fans to get behind the next boss.
"The new manager will have to have a big personality," he said. "They are big boots that Malky has left. But I am sure the owner and the board of directors will select correctly.
"It is certainly going to take them into a new direction and the supporters need to get on board and support whatever they do."
Taking a side to the Emirates in the Premier League should rank as one of the biggest days in Kerslake's career but he admits it will feel slightly surreal as he considers Cardiff to still be Mackay's side.
He said: "I don't feel it is my side. I am going there and doing the job the club have asked me to do. Of course it is the Premier League, a great stadium and something I am looking forward to."
Cardiff will at least avoid coming up against Aaron Ramsey. The Wales midfielder scored twice in a 3-0 Arsenal win when the sides met in Wales but has been ruled out with a thigh problem.
Not that Kerslake expects it to have an advserse effect on the Gunners.
"They could field three teams with world-class individuals so I don't think that will be any challenge to their manager," he said.
"But we have gone up against top sides before and got a result and we will take a game plan and we will stick to it and we will be okay."