Carrick is currently contracted to June, though has guided the club into the top three.
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Sheringham said on talkSPORT: "Michael Carrick's gone in there, he's done an unbelievable job.
"And it just shows you what can be done if you've got the right approach, if you get on the right wavelength with players, things turn around very, very quickly.
"So he's gone in there, he's had that reaction from him, and they've gone on from there. I'd be very disappointed in Man United if they didn't give Michael Carrick the job.
"I think he's been fantastic since he's gone in there."