The 19-year-old Croatian has spent the season on loan at Hamburg, where he established himself as one of the most highly-rated young centre-backs in the Bundesliga.
Vuskovic joined Tottenham from Hajduk Split in 2023 as a teenager, before being sent to Hamburg on loan to continue his development.
He went on to make 30 appearances across all competitions for the German club this season.
His performances earned him a place in Croatia's squad for the World Cup, where he could feature against England in the group stage.
Brighton's £30m bid was described by sources as an attempt to test Tottenham's resolve over a player the north London club rate far more highly.
Spurs are understood to value Vuskovic at more than £60m, a figure more than double Brighton's opening offer.
The interest from Brighton comes amid wider talks between the two clubs over Dutch defender Jan Paul van Hecke, who has been the subject of previous bids from Tottenham.
Brighton's chief executive Paul Barber confirmed last weekend that two bids from Spurs for Van Hecke had already been rejected.
Vuskovic's situation is complicated by his lack of senior involvement at Tottenham since his arrival from Croatia.
Despite that, the club are understood to want to tie him down to a new contract rather than sell, viewing him as part of their long-term defensive plans.
Tottenham have also been busy elsewhere at centre-back this summer.
The club have already agreed the signing of Marcos Senesi on a free transfer, with the move to be completed once his contract at Bournemouth expires on July 1.
That depth at the back makes Vuskovic's path to regular first-team football at Tottenham less certain, even with his stock having risen sharply over the past year.
Reports in Germany have also linked Barcelona and Bayern Munich with the defender, reflecting the scale of interest generated by his loan spell.
For now, indications are that Tottenham will wait until after the World Cup before making any final decision on his future, with his performances in the United States likely to factor into that judgement.
Whatever happens, Vuskovic's rise from a teenage arrival with no senior appearances to a player valued at £60m within two years marks one of the more dramatic value shifts of the season.