The 44-year-old took over for the rest of the campaign in January following an acrimonious end to Ruben Amorim's rocky Red Devils reign.
United clinched Champions League qualification with three matches to spare after being out of the competition for two seasons.
Carrick, appointed on January 13th, has won 10 of his 15 matches in charge in all competitions, losing just twice, to guide United from seventh to third in the Premier League table.
The Athletic reported that chief executive Omar Berrada and director of football Jason Wilcox would recommend to co-owner Jim Ratcliffe that Carrick be offered the role at an executive meeting this week.
United have explored other options but Carrick has long been considered the frontrunner for a job that he has been publicly backed for by a number of players. Some bookmakers have now suspended betting on his appointment.
Carrick is set to face the media on Friday ahead of Sunday's final home game of the season against Nottingham Forest.
