After Stephen Kenny's departure, the FAI enter interviews with a series of major coaching names - including Solskjaer and his former Manchester United captain Roy Keane.
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Ultimately, Heimir Hallgrimsson was named Republic of Ireland coach.
Solskjaer recalled this week: "I had some good conversations.
"But, in the end, it didn't turn out that way and I'm very happy where I am now, we'll put it that way."
Besiktas meet St Patrick's Athletic, with Kenny in charge, tonight in a Europa Conference League qualifier.