The offer, understood to be £42m plus £3m in add-ons, follows two previous bids worth up to £35m and still falls short of Tottenham's valuation for the Croatia international.
Vuskovic, who spent last season on loan at Hamburg, is keen to play regularly and is open to linking up with Brighton manager Fabian Hurzeler for the chance of consistent Premier League football. He has been capped six times by Croatia, starting their opening World Cup defeat by England, and is under contract at Tottenham until 2030.
Brighton's pursuit comes as the club works to rebuild its centre-back options following a busy summer of departures. Adam Webster left on the expiry of his contract, while Van Hecke's £52m move to Tottenham was completed earlier this week after the Dutchman had just a year remaining on his deal.
To further bolster their defence, Brighton have also opened talks for Venezia's Michael Svoboda, a 27-year-old Austrian centre-back.
Brighton have separately confirmed the signing of striker George Munday from Cambridge United on a two-year contract for undisclosed terms, with the 19-year-old joining the club's Under-21 squad as they enter the new season as reigning Premier League 2 champions.
Under-21s head coach Shannon Ruth welcomed the signing.
"George had an excellent second half of the season with Hereford, having previously made his professional debut for Cambridge United," Ruth said. "We're looking forward to bringing him into our environment and seeing his game continue to develop."
Munday joined Cambridge's academy at the age of nine and made his first-team debut in 2024, before loan spells at St Ives Town and Bedford Town. He found his scoring touch at National League North side Hereford last season, netting nine goals in 22 appearances.
The two moves reflect Brighton's parallel approach to the summer window — strengthening the first-team defence with experienced Premier League-ready options while continuing to invest in promising young attacking talent for the development squad.