He is concerned that his limited role at United may affect his chances of being included in the Dutch national team's World Cup squad.
Roma are interested in signing Zirkzee on loan, as the club's finances are tight.
Roma's coach, Gian Piero Gasperini, is a fan of Zirkzee's abilities, but competition from West Ham, who are also monitoring the situation, may arise in the second half of the season.