According to Italian journalist Niccolo Ceccarini, the Serie A champions are seeking midfield reinforcements amid uncertainty surrounding Andre-Frank Zambo Anguissa, Stanislav Lobotka and Kevin De Bruyne.
Bergvall, who attracted interest from Italian clubs during the summer, could become available when the transfer window reopens.
The youngster arrived at Tottenham with considerable promise but has struggled to establish himself as a regular under Roberto De Zerbi.
Napoli believe the midfielder could strengthen their squad, although competition from Barcelona may complicate negotiations.
A move to Italy could offer Bergvall greater opportunities for regular first-team football and help revive his development after a frustrating spell in the Premier League.