The left-back once again starred for Italy in their 3-0 win over Switzerland at Euro 2020 on Wednesday.
It followed on from his man-of-the-match performance against Turkey.
The Expresssays the two London clubs are monitoring Spinazzola's performances.
But they could face competition from Tottenham, who are expected to announce Paulo Fonseca as their new manager in the coming days.
Fonseca worked alongside Spinazzola at Roma for two seasons.