The 20-year-old remained at Tottenham despite receiving interest during the summer, when Nottingham Forest and Newcastle United reportedly submitted substantial offers for him.
Chelsea, Manchester United, Borussia Dortmund and Bayern Munich were also linked with the Sweden international, but Tottenham rejected the approaches and kept Bergvall in their squad.
He subsequently signed a new agreement around a month ago, although the deal was an adjustment rather than an extension. His contract still runs until the summer of 2031.
The new contract has not, however, removed uncertainty over his immediate future.
Tuttomercatoweb reports that Bergvall could consider leaving Tottenham in January because of the intense competition for midfield places.
Bergvall has struggled for regular minutes under Roberto De Zerbi this season, having made only three Premier League appearances so far. He played 45 minutes against Brentford, 27 minutes against Everton and six minutes against Aston Villa, while also starting Tottenham's EFL Cup matches.
The lack of playing time could become an issue for the midfielder, particularly with his position in the Sweden squad also important to his long-term ambitions.
Serie A provides several potential destinations. Fiorentina sporting director Fabio Paratici knows Bergvall from his time at Tottenham and is reportedly interested in bringing the midfielder to Florence.
Juventus are also looking for a player with Bergvall's characteristics, while Milan and Inter could consider him as a longer-term midfield option.
Napoli have also been identified as a possible destination if they lose one of their prominent players.
Tottenham therefore face a difficult decision despite having recently secured Bergvall on improved terms. The club can point to a contract that runs until 2031, but the midfielder's lack of opportunities could force the issue if January approaches without a significant change in his role.
For Bergvall, the attraction of Serie A would be the prospect of regular football. For Tottenham, allowing a player they resisted selling during the summer to leave only months after renewing his agreement would represent a significant change in direction.