The German is one of European coaching's hottest young properties and is available, having left Borussia Dortmund in May.
But Tuchel is understood to be on the lookout for a Champions League club with which to resume his career, and is happy to wait for the right opportunity to come along.
Frank De Boer is heavily fancied by the bookmakers but has had no talks with the club, even though his agent Guido Albers brought the immensely successful Ronald Koeman to St Mary's.
Mauricio Pellegrino, the favourite to take over at Crystal Palace, is one name that definitely has piqued Southampton's interest, and the club are understood to have been sounding out potential candidates for weeks as sporting director Les Reed leads the hunt for a new coach.
Reed has a reputation for sniffing out talented young managers and is being trusted to bring in a new coach who can get Southampton playing again after criticism of Puel's style.Reuse content