Malky Mackay was sacked by Cardiff on Friday and former Manchester United striker Solskjaer was immediately installed as the favourite to take charge.
Solskjaer is currently coach of Norwegian side Molde and has regularly been linked with a return to English football over the past 18 months.
Dalman has spoken to the person who is his No.1 choice to succeed Mackay, but was coy about reports that Solskjaer will be handed the Cardiff City Stadium post.
"I have a preferred candidate. I only have one candidate in mind," said Dalman at the post-match press conference following Cardiff's 2-2 draw with Sunderland.
"I have not made an offer to anybody as such, but that does not mean I have not had discussions."
When asked if he had spoken to Solskjaer, he said: "I spoke to him many a time in the old days (when the Norwegian was at Manchester United).
"I might have spoken to him recently, and I might not have."
He added: "You know what these things are like, they take time. We will do our work as diligently and as quickly as we can but I am reasonably optimistic that we will get things moving."
One man Dalman ruled out of contention is Turkish journeyman Vural, whose agent has consistently touted him for the Cardiff job.
But Dalman said: "I am not getting a Turkish manager, I can categorically state. There is a manager who keeps putting himself forward. It is not going to happen."
Dalman confirmed Mackay's lieutenants David Kerslake and Joe McBride will remain in temporary charge of the team against Arsenal on New Year's Day.
Kerslake made it clear he has no aspirations to take the full-time job, commenting: "I have been in the position for six hours and had three phone calls from the director, one from the chief executive, one from the chairman and one from the owner. That's probably put me off for life!
"They were all well wishes as well, by the way, but I have been a coach for 10 years. I love what I do. I love being on the grass with the players."