Spurs had two previous offers rejected before completing the agreement for the Netherlands international, who had been set on a reunion with Roberto de Zerbi following the Italian's move to the south coast.
Any negotiations over Luka Vuskovic, the subject of two rejected Brighton bids worth up to £35m, remain entirely separate from the Van Hecke deal.
Vuskovic is keen to secure regular football after a loan spell at Hamburg, and has also attracted interest from Barcelona.
Tottenham have also targeted Newcastle midfielder Sandro Tonali as they look to build a squad capable of competing for Champions League places again.
De Zerbi is hoping Van Hecke will become his third defensive signing of the summer, following free transfer deals for Andy Robertson from Liverpool and Marcos Senesi from Bournemouth.
The move comes amid a major rebuild at Tottenham following a campaign in which the club only avoided relegation on the final day of the season.
Van Hecke played in the Netherlands' 2-2 draw with Japan at the World Cup, and is expected to add depth to Tottenham's defence as the club prepares to sell captain Cristian Romero.
The 26-year-old joined Brighton from NAC Breda in 2020 and had just a year left on his contract at the Amex Stadium.
Brighton chief executive Paul Barber had previously insisted the club would hold firm on their valuation.
"There's always going to be a lot of interest in our best players, and certainly in the case of Jan Paul.
"We've been very clear that that interest has been ongoing for a while, it's coming from multiple sources.
"Yes, we have rejected a bid from Tottenham over the last week or so, in fact, two bids. From that point of view, it has to be right for us as well as the player.
"We have to be in a position to make the best trades to suit our model and also to make sure that we're supporting Fabian, because he's got another big season ahead of him."