The Real Madrid defender is out of contract at the end of the season and could be snapped up for free.
His wages would perhaps be the biggest stumbling block, but they would save millions with no transfer fee.
Bruno Fernandes plays almost every game for United so they could look to provide the Portugal international with back-up.
Donny van de Beek was brought in from Ajax last summer to do just that but Solskjaer does not seem the rate the Dutchman.
Jesse Lingard is expected to be sold, with West Ham and Arsenal hot on his tail.
Eriksen has failed to make the grade at the San Siro following his move from Tottenham and could add some creativity which may take United to the next level.