United were strongly linked with Real Madrid's Gareth Bale during the summer transfer window, but the Welshman now looks certain to stay at the Bernabeu for the long term.
Neymar's future at PSG is less secure, however, with the Brazil international believed to be open to leaving the club he joined from Barcelona in the summer of 2017.
The forward, who is the most expensive player in world football, has scored 32 times in 36 appearances for PSG since his arrival from Camp Nou.
Madrid have long been credited with an interest, but according to Don Balon, United have made Neymar their top target in 2019, with head coach Mourinho desperate to secure his services.
Neymar managed 28 goals in 30 appearances for PSG last season before a broken foot ended his campaign, but he recovered to represent Brazil at the 2018 World Cup.