The French centre-half has committed to new terms to 2030.
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L'Equipe says Burnley moved quickly to upgrade Esteve's contract after learning of Tottenham's interest. An announcement is due from the Clarets in the coming hours.
Arriving from Montpellier last year, Esteve, 23, made an immediate impact on English football.
With 46 Championship appearances, he was a cornerstone of the Clarets ' defence, who finished the season with just 16 goals conceded, a league record.
His performances attracted the attention of several Premier League clubs, including Crystal Palace and especially Tottenham, but Burnley were adamant: Esteve is not for sale.