Carrick has given himself a real chance of earning an extended stay, but he can guarantee it by bringing the Premier League title back to Old Trafford. United are still 12 points behind current leaders Arsenal, but Mikel Arteta's side have only picked up two points from their last three games, and the nerves that derailed their previous title bids appear to be setting in.
Arsenal don't have the swagger of champions; they have become far too reliant on set-pieces and their frontline is suffering in open play because of it. Second-placed City are enduring similar struggles in the final third amid a worrying injury crisis, while no one expects Unai Emery's over-achieving Aston Villa to stay the course in third.
All three of those clubs are having to juggle commitments across multiple competitions, too, which gives United a clear advantage. Carrick only has 15 games left to prepare for, and in theory, his team should be fresher than the majority of their opponents.
Aside from home matches against Aston Villa and Liverpool and a trip to Chelsea, their fixture schedule is also favourable. Another win against Fulham at Old Trafford next weekend could see United emerge as dark horses, with Arsenal facing a tricky trip to an improving Leeds side and City traveling to Tottenham, who have beaten them in three of their last four meetings.
Perhaps privately, Carrick will be taking inspiration from United's class of 1996, who were 12 points adrift of Newcastle after 23 games, but pulled off a miraculous turnaround to clinch the title by four points. Champions League qualification should only be the minimum aim for the Red Devils now that they are back in the top four; they have the quality to push for the ultimate prize.
Although Arsenal are still favourites because of their superior strength in depth, United are gathering momentum. If they can keep up their newfound level and adapt when needed against the so-called lesser teams that play with a low block, anything is possible. Carrick's at the wheel, and so far, it just feels right.