Brazilian midfielder Antony, Dutch defender Tyrell Malacia and England international Jadon Sancho are also looking to leave the club, the source added.
Rashford, who has fallen out of favour with manager Ruben Amorim, played for Aston Villa on loan while United had their worst-ever Premier League campaign, finishing 15th in the standings.
Sancho was on loan at Chelsea, while Antony was at Real Betis and Malacia played for Dutch side PSV Eindhoven.