Old rival Joe Hart and Celtic goalkeeper Craig Gordon are both on Manchester United's wishlist according to reports.
Hart has been at Manchester City for more than a decade but has spent the last two years away from the Etihad on loan, first in Italy with Torino before joining West Ham last summer.
The Sun reports Mourinho wants to take the former England number one across Manchester as back-up to first-choice goalkeeper David de Gea.
If that deal does not materialise, The Sun also reports Gordon could make his way to Old Trafford instead.
The 35-year-old has a contract with Celtic until 2020, but may trade his role north of the border to join the Premier League runners-up.
One player who could be on their way out of Old Trafford is Marouane Fellaini, with the Daily Mail reporting he has already held negotiations with Marseille.
But Fellaini, the paper states, is waiting on an offer from Manchester United to renew his contract, and favours a move to AC Milan.