News broke after their 3-2 victory over Arsenal that Carrick, who was expected to assist interim head coach Ralf Rangnick, would leave Old Trafford after 15 years.
The 40-year-old was placed in caretaker control of the first team after Ole Gunnar Solskjaer's sacking in November.
He remained unbeaten in his three games in charge and United impressed with their performances.
The Telegraph says the Old Trafford hierarchy, which includes the Glazer family, could turn to the former midfielder in any future search for a new manager.
Carrick's departure ended his long association with the 20-times English champions, having first signed for the club in 2006 before taking on a coaching role after his retirement.
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