The Italians failed to qualify for the 2018 World Cup last summer, but have enjoyed a strong start to Euro 2020 qualification under Mancini, winning both of their opening games against Finland and Lichtenstein.
Their qualification campaign continues against Greece on June 8, before Italy host Bosnia-Herzegovina three days later.
"If we do good work, if we play attacking football that is entertaining and gratifying, then results are possible," Mancini toldLa Stampanewspaper.
"We are Italy, we can be like France and Spain in opening up a new era. It's a good moment in our football.
"If we were to face France now, they’d beat us, but we’ll be able to have our say once Euro 2020 comes around.
"My idea was originally to target success in the 2022 World Cup, but now we can challenge for Euro 2020."
Mancini has introduced a number of emerging talents into Italy's squad since taking over in May last year, and though he acknowledged inexperience could prove to be a problem, he has no concerns over throwing youngsters such as Moise Kean and Roma star Nicolo Zaniolo in at the deep end.
"We have some real young talents coming through," Mancini added.
"Kean is only 19 years old, so it's up to him, but he can become an extraordinary striker. I sometimes use him on the wing, but if he does what he needs to, then he’ll be a centre-forward.
"Then there are Sandro Tonali, Nicolo Barella, Gianluca Mancini and Zaniolo. They look pretty talented to me and also physically strong, so we can have our say.
"There is the issue of experience, because in my day, we'd reach their age and already have three years of Serie A football under our belts.
"On the other hand, it's refreshing to see their reactions and naivety."