Spanish outlet El Pais reported the deal for around 400 million euros covers 80% of the club's total shares at an estimated 3,500 euros per share.
The transaction requires approval from LaLiga and the Spanish National Sports Council before it can be completed.
Sevilla have been ever-present in LaLiga since the 2001/02 campaign, but the Andalusian club are fighting relegation this season after a poor campaign.
The club, who won the Europa League in 2022/23, are 13th in the standings but sit only three points above the relegation zone with three games left. They finished 17th last season, one point above the drop zone.
Ramos, who won the World Cup with Spain and two European Championships, began his youth career at Sevilla before joining Real Madrid in 2005 and helping them to win 22 major trophies.
The 40-year-old returned to Sevilla in 2023 for a season and last played for Mexican club Monterrey.