Premier League United sacked manager Jose Mourinho on Tuesday after having suffered their worst start to a season for 28 years.
Although United would neither confirm nor deny media reports about who would take charge, the source said Solskjaer, who scored the goal that won United the treble in 1999, and Phelan were "clear front runners" and a decision would be made shortly.
Solskjaer, 45, played for United between 1996 and 2007 and is currently manager of Norwegian club Molde.
Phelan played for United under former manager Alex Ferguson between 1989 and 1994 and worked as a coach at Old Trafford for 14 years.
Both Solskjaer and Phelan bring a strong connection with United's illustrious past.
Solskjaer was the hero on one of the club's most famous nights, scoring the winner deep in stoppage time as United completed a 2-1 comeback triumph over Bayern Munich in the 1999 Champions League final to complete an unprecedented treble.
He was largely a 'super-sub' at the club, a reliable back-up who still managed to find the net 126 times.
United's next game is at Cardiff City on Saturday, where Solskjaer was manager for nine months in 2014.
Phelan worked in various coaching roles at Old Trafford but became Ferguson's assistant in 2008 and was his right-hand man during three Premier League title-winning campaigns.
(Writing by Toby Davis; Editing by Ken Ferris)
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