The Italian was able to assess James up close and personal when Hungary secured a surprise victory over Wales in a Euro2020 qualifier last month.
It was a match also watched closely by United manager Ole Gunnar Solskajer and transfers chief Ed Woodward.
The Red Devils made their move for James towards the end of June and eventually completed a deal earlier this month, paying Swansea City an initial fee of £15m that could rise to £18m with add-ons.
21-year-old James has impressed so far in United's preseason tour of Australia, and Rossi thinks there is only more to come from the former Hull City winger.
Speaking exclusively to tribalfootball.com, Rossi said: "I believe that James can only grow.
"He is young and has excellent qualities, with explosiveness and speed.
"And training daily with United will give him the chance to improve in the individual tactical aspect within a short time."
James started as a forward for Ryan Giggs' Wales side in the loss against Hungary in Budapest.
But Rossi admits James worried his side with his blistering pace.
He added: "In the June game in Budapest we tried to limit him, I think we succeeded.
"Even in those 2/3 situations in which he escaped he created us a lot of apprehension."