Amorim, 40, was relieved of his duties on Monday (January 5), following his outburst in his post-match press conference after the 1-1 Premier League draw with Leeds.
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Former midfielder and academy coach Darren Fletcher will take charge of their game against Burnley on Wednesday but may not see out the rest of the season.
According to Fabrizio Romano, Solskjaer is keen on taking the job regardless of the contract length and is in talks with the club.
The 52-year-old took a similar pathway back in 2018, when he was handed temporary charge after Jose Mourinho was sacked before getting the job on a permanent basis.