The Magpies are in the market for a new midfielder after Tonali completed a club-record £100 million move to Spurs earlier this month.
Talks have taken place between the two clubs, with Tottenham understood to be open to a sale only if their valuation is met, according to Sky Sport.
Nottingham Forest have also failed with a £38 million bid for the 20-year-old Sweden international, who remains one of their priority targets.
Bergvall is reportedly keen to leave Tottenham in pursuit of regular first-team football in his preferred box-to-box midfield role.
Despite Spurs' reluctance to sell, interest in the highly rated youngster continues to grow.
