Speaking before Friday's pre-Euro friendly with San Marino, Mancini said that Paris Saint-Germain midfielder Verratti is recovering from the knock "quicker than expected".
"His situation has improved a lot in the last 15 days," said Mancini, who has until June 1 to selected his final 26-man squad for this summer's tournament.
PSG had ruled the 28-year-old out for four to six weeks a fortnight ago, which in the worst case would mean him missing Italy's three Group A matches with Turkey, Switzerland and Wales.
On Tuesday, Italy's team doctor Andrea Ferretti said that Verratti's knee was "not bad" and hinted at him entering the fray with the tournament underway.
Other members of Mancini's provisional 33-man squad carrying knocks are defender Leonardo Spinazzola and midfielder Lorenzo Pellegrini of Roma, Lazio defender Francesco Acerbi and Inter Milan midfielder Stefano Sensi.