Guardiola, who left Manchester City at the end of last season, is one of several names under consideration, with Antonio Conte, Roberto Mancini and Andrea Pirlo also having been linked to the position.
Italian Football Federation president Giovanni Malago confirmed the contact with Guardiola, telling Cronache di Spogliatoio, "I think it was right and important to start a dialogue and keep it alive."
Gattuso departed after Italy lost their World Cup play-off to Bosnia and Herzegovina in March, extending one of the most difficult periods in the country's football history after missing out on three of the last four tournaments.
Malago was careful to stress that Guardiola's candidacy does not mean other options have been shelved.
"It is absolutely not about a lack of respect or consideration for other candidates, since discussions have already begun about other names, and then you can choose someone who perhaps belongs to a newer generational school, so to speak, and the question there is highly subjective," he said.
Reports last week suggested Guardiola's salary demands could prove an obstacle to any national team appointment, given the financial constraints typically associated with international football.
Malago addressed that directly, saying: "Exceptions have been made, exceptions that for example concern the name that is so dominant right now: Pep Guardiola. Exceptions for obvious reasons that I won't go into here, but it isn't certain that this (the appointment) will happen."
Guardiola won six league titles in ten years at Manchester City before leaving the club this summer, along with major honours across spells at Barcelona and Bayern Munich, and would represent one of the most eye-catching appointments in international management if he does take charge of the Azzurri.
The 55-year-old has been out of work since ending his decade-long reign at the Etihad Stadium, and a return to club football has not been ruled out either, with Italy's federation understood to be racing against other suitors to convince him that an international project is the right next step in his career.