The Holland international is expected to leave United in January and is attracting interest from West Ham and Everton.
However, the Mirror says Roma have made a first formal move by approaching United about Zirkzee's availability.
Talks are also underway with the former Bologna star's agents.
However, Roma face stiff competition, as along with the Irons and Everton, PSV Eindhoven and Serie A rivals Juventus are also keen.
Roma hope to set up a loan to buy arrangement, with the permanent option dependent on appearances and if they qualify for Europe.