Ramos, 37, has been a free agent since his Paris Saint-Germain contract expired in July.
Club president Pepe Castro appeared to rule out a move for Ramos last month, saying he had "never been an option" and said the defender had been offered to the club "directly and indirectly."
Sevilla -- who won the Europa League in May -- have had a poor start to this LaLiga season, losing all three matches so far and are bottom of the table.
Ramos was born in Camas, a town just outside the city of Seville, and played for Sevilla as a teenager before joining Real Madrid in September 2005.
He spent 16 years at Madrid, becoming club captain and winning five LaLiga titles and four Champions Leagues -- providing one of the most iconic moments in the club's history with his goal in the 2014 Champions League final -- before joining PSG on a free transfer in July 2021.
After an injury-hit first season in Paris and a more impressive second campaign, Ramos left PSG this summer and has since been considering his options.
The former Spain captain retired from international football in February -- two years after his last appearance -- when he was told he was not part of coach Luis de la Fuente's plans.
Ramos was a key part of the Spain teams which won three back-to-back international tournaments between 2008 and 2012, including the 2010 World Cup.