The Turkish newspaper AMK states that Muslera is keen on a move away from Galatasaray at the end of the season.
The report goes onto states that the Red Devils will offer €15 million for the Uruguay international.
Galatasaray have been banned from European competition next season due to failing to meet Financial Fair Play regulations.
The Istanbul giants have announced that they will consider offers for their star players over the summer transfer window.
Muslera was also linked with a move to Manchester Untied over the summer before David de Gea signed a contract extension.
The 29-year-old played a key role in Galatasaray winning the league for a record 20th time last season and earning the right to wear a fourth star – the Lions also won the double for a record sixth time.