Mackay was relieved of his duties on Friday, 11 days after owner Vincent Tan had sent him an ultimatum by email to resign or be sacked.
Solskjaer was immediately installed as the bookmakers' favourite to take the reins at Cardiff City Stadium and, although Press Association Sport understands that no formal offer has yet been made to the former Manchester United striker and current Molde manager, he is firmly on the Bluebirds' radar.
Just last week Molde confirmed they had received approaches for Solskjaer, but would not reveal from which clubs.
It is also understood that the next Bluebirds' boss will be backed by Tan in the transfer window; Mackay was told just last week "not a single penny" would be available to him in January.
Former England boss Sven-Goran Eriksson, Turkish journeyman Yilmaz Vural and Basle boss Murat Yakin are also believed to be among the contenders.
Cardiff's honorary life president and former owner Sam Hammam admitted Solksjaer fitted the profile the club would be looking for, and also played down the possibility of recruiting Eriksson.
He told BBC Wales: "He (Solskjaer) will have had very good basic grounding (at Molde) to what is needed now.
"We need a manager whose feet are on the ground and not just telling Vincent Tan, 'Give me your money,' as if all Vincent Tan is is a cash cow."
Mackay's exit came as no surprise after the complete breakdown of his relationship with Tan in recent months.
The rift became public knowledge when Iain Moody, Mackay's head of recruitment, was dismissed in October over an alleged transfer window overspend.
Tan was then said to be "extremely upset" when Mackay suggested he would like to add three players to his squad in January, and sent the 41-year-old his ultimatum.
Chairman Mehmet Dalman looked to have earned Mackay a reprieve when he released a statement saying he would remain in charge for the "forseeable future".
But peace talks between Tan and Mackay never materialised, and the 3-0 Boxing Day defeat to Southampton proved the latter's final game in charge.
Assistant manager David Kerslake and first-team coach Joe McBride will take charge of Saturday's fixture against Sunderland.
Mackay's dismissal was confirmed in a short statement on the Cardiff website, which read: "The board of directors at Cardiff
City Football Club have today relieved Malky Mackay of his duties.
''A new first-team manager will be appointed and announced in due course.''
Mackay responded with a statement, and believes his accomplishments in leading the club to the 2012 League Cup final, the 2013 Championship title and a solid start to their maiden Premier League campaign, give him reason to be proud of his time in the Welsh capital.
He said: "I leave with my head held high having gained a level of experience that, upon reflection, I suspect would have been difficult to find anywhere else in British football.
"The foundations laid at Cardiff City will provide a solid platform for my successor, and for the sake of the club, I hope that all of the progress that has been made so far will not be impaired in any way by today's enforced changes."
He added: "As disappointing as today's decision is, I am already able to look back at my time at Cardiff City FC with immense pride and joy given the number of milestones the club achieved during the last two and a half years.
"A League Cup Final appearance followed by promotion to the Barclays Premier League for the very first time, achieved as winners of the Championship, allowed the club a move back to this elite level after a 51-year absence."
Tan claimed the decision to dispense with the services of Mackay was down to the Scot airing the club's ''dirty linen'' in public.
''There has been a good deal of publicity generated by, and about, Mr Malky Mackay for the last few months,'' he said in a statement.
''Indeed, far too much dirty linen has been exposed to the public gaze but, I stress, not by me.
''Indeed, I have deliberately not responded to this, hoping that the club can be judged on its football rather than personalised arguments about who said what to whom.
''I have, however, regretfully concluded that it is no longer fair to the club, its players, its fans and the public more generally for this uncomfortable state of affairs to continue."
League Managers' Association chief executive Richard Bevan spoke of his disappointment at Mackay's departure, but praised the Scot for the manner in which he had conducted himself during a difficult period.