Allegri and Barzagli were teammates at Pistoiese in 2000 when Allegri noticed that his colleague was not playing to his strengths.
Inter have made an offer for 16-year-old Pellegri, who scored his first Serie A goal on the final day of last season, and 15-year-old Salcedo.
Totti brought an end to his 25-year career with Roma at the end of the season.
Roma have completed the signing of full-back Rick Karsdorp from Feyenoord, with the 22-year-old signing a five-year contract with the Giallorossi.
Reports in the Italian media suggested he will complete a £5.3 million permanent transfer.
Former footballer Paolo Maldini's professional tennis debut ended in defeat at the Aspria Tennis Cup in Milan.