Bergvall has featured heavily across his first two Premier League seasons, but his role diminished significantly after Roberto De Zerbi took over as manager in the spring.
He managed one goal and five assists in 33 competitive appearances last season, 17 of them starts, but has since found himself squeezed further out of the picture by Tottenham's big-money summer additions of Sandro Tonali and Mateus Fernandes.
The Athletic reported in June that Bergvall had informed the club's hierarchy of his desire to move on in search of regular first-team football. De Zerbi has said he would be happy to keep the Swede, but has been clear that he only wants players in his squad who genuinely want to be at the club.
Tottenham have already turned down approaches from both Newcastle and Nottingham Forest.
According to the Daily Mail, Newcastle United remain interested, with Tottenham continuing to demand £50 million to sanction a sale.
A move would allow Bergvall to effectively replace Sandro Tonali in Newcastle's midfield plans, with Tonali himself having strengthened Tottenham, while Bruno Guimarães has separately been sold to Arsenal.
Nottingham Forest are not out of the running either, with the paper reporting the Midlands club are expected to return with a fresh offer, having previously seen a bid of around £46 million rejected alongside Newcastle's own approach at a similar level.