The 49-year-old left Galatasaray at the end of the season after just nine months in charge, despite guiding the club to a second-place finish in the Super Lig.
Cesare Prandelli resigned as Italy coach following their group-stage exit from the World Cup in Brazil, which has led to reports suggesting that Mancini will replace him.
"Italy would be an honour for me," Mancini told Corriere dello Sport.
"No-one called me but I would be proud. [The] FIGC (Italian Football Association) need serious and competent men."
Prandelli was appointed as Mancini's successor at Galatasaray on Monday.