The Holland striker, formerly of Bologna, is being linked with a return to Serie A before Tuesday's winter market deadline.
Roma and Juventus are both keen to setup loan-to-buy deals for Zirkzee.
But The Sun says Zirkzee's agents have informed management that he wants to stay and play for manager Michael Carrick.
And the feeling's mutual, with Carrick saying before Sunday's visit of Fulham: “I think Josh has just been trying to work to get fit and get back around the group and it’s been great to have him back training yesterday with a group for the first time.
“That’s literally as far as it goes for me and for Josh. I’m looking forward to working with him. That’s it.
“Nothing else needs to be said.”
