Bruce is under pressure at the Stadium of Light after a large section of the club’s support turned on him following the 2-1 home defeat by Wigan last weekend.
Solskjaer has been sounded out for Championship jobs, but has shunned the approaches because he would rather stay in Norway and lead Molde into Europe.
But reports claim Solskjaer would consider leaving Molde for a Premier League job.
Solskjaer helped his current club Molde to the first league title in the history of the club in his first season in charge this year.
Solskjaer also has experience as a manager of the Manchester United’s reserves.
Bruce’s former Manchester United team-mate Mark Hughes has also been linked with the Sunderland job.