Liverpool were also interested with recent reports claiming that the 27-year-old already told his friends that he will leave the St Mary's Stadium this summer.
Antonio Conte's men will, however, face fierce competition for his signing with Manchester City and Liverpool also believed to be vying for his services.
Ederson, who has become the most expensive goalkeeper in history, will officially join Man City on July 1.
City boss Pep Guardiola has been tracking Bertrand for months and could move for the England worldwide if he fails in his pursuit of Monaco left-back Benjamin Mendy. He is putting in place a young team for the future.
With Virgil van Dijk's "will he, won't he" transfer saga dominating Southampton's headlines since the season finished, another key Saints player has seemingly slipped through the club's fingers, as Ryan Bertrand looks set to make a shock switch to Manchester City. Those were important factors in making my decision. "He is young, has all the qualities you could want as a top-class goalkeeper and I think he'll fit in really well to our squad".
"He's important for us and now, of course, we will wait to the beginning of the season to see him on the pitch with Southampton".
Meanwhile, director of football Txiki Begiristain said City have signed one of the best in his position.