Bergvall, 20, is understood to want the move to Nottingham Forest, having already informed Tottenham directly of his desire to leave this summer.
Tottenham have strengthened their own midfield significantly this window, bringing in both Sandro Tonali from Newcastle and Mateus Fernandes from West Ham, moves that have contributed to uncertainty over Bergvall's future at the club given the increased competition for places.
The Sweden international's camp made his intentions clear at the end of June, when The Athletic reported he had told Tottenham he wanted to depart, with Forest emerging as his primary suitor after selling Elliot Anderson to Manchester City for a reported £116m.
Tottenham are reported to value Bergvall at around £50m, having already knocked back the Midlands club's initial approach earlier in the window.
The Telegraph and talkSPORT both reported earlier in the day that Forest were intensifying their pursuit, and BBC Sport's figure of £45m would represent a significant improvement on the rejected opening bid, even if it still falls short of Tottenham's stated valuation.
Forest, now under the management of Oliver Glasner, are keen to find a direct midfield replacement for Anderson and view Bergvall as an ideal fit for that role given his age, technical quality and ball-carrying qualities from midfield.
Bergvall, who joined Tottenham from Djurgården ahead of the 2024-25 season and signed a contract running until the summer of 2031, made one goal and three assists in 23 league appearances last season, having been used more sparingly than in his debut campaign in English football.
Despite his contract security, the midfielder's determination to secure regular first-team football appears to be driving the push for an exit, with Forest continuing to negotiate patiently in the hope of eventually meeting Tottenham's asking price and completing a deal before the transfer window closes.