Gattuso, 35, sacked earlier this season as coach of second division Palermo, is suspected of "associating with criminals with intent to commit sporting fraud".
The long-running "Calcioscommesse" saga took a further twist with the arrest of the four, all said to have provided the link between players and a clandestine betting ring.
Police declined to say where the arrests had taken place, but the nationwide operation was led by the Cremona prosecutor's office, which initially uncovered the scandal in 2010.
Spadaro is said to be linked with the "Bolognese" gang which was involved with former player Giuseppe Signori, who was arreted in 2010 in the intial investigation into match-fixing.
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Bazzani has also been linked with Signori, as well as Almir Gegic, the alleged ringleaser of another match-fixing gang.
Rinci, chairman of the Riccione football club, and Quadri both have links with Spadaro.
Gattuso, renowned as a tough-tackling midfielder who spent the best years of his career with AC Milan from 1999 to 2012, is not the only former player suspected of involvement.