The current Arsenal midfielder was at West Ham United at the time, with United linked to his signature over several summers.
However, no move materialized, with United spending money on the likes of Donny van de Beek, Amad Diallo, and Facundo Pellistri instead.
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Rice wanted to sign for United back in 2020 and 2021, but Solskjaer admits they did not push the boat out to sign him.
Speaking on the Stick to Football podcast, Solskjaer said: "I really like him (Rice) as a player.
We've discussed him a few times and I think we would've done well with him in midfield.
His legs, covering the pitch, and I think he improved a lot on his playmaking as well.
Obviously, he would've cost some money, but we could've and should've gone for him."