According to The Athletic, Saints have not yet accepted the £15m offer and would only sell the Denmark international for £18m.
Vestergaard, who only has a year left remaning on his Saints contract, joined Southampton from Borussia Monchengladbach for a £22m+ fee in the 2018 summer transfer window.
The towering Dane played a bit-part role in Southampton’s defence up until last season, where he had a breakout campaign for the Saints and was easily one of Ralph Hasenhuttl’s best players over the last year.
He also impressed while representing his country at EURO 2020 this summer and joined up with his Southampton teammates in pre-season training last week after a mini post-tournament break.
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With a year remaining on his current deal, Hasenhuttl last night suggested that he would like Vestergaard to stay but the club need to be smart in a situation similar to what saw Danny Ings leave for Aston Villa last week.
“It’s never an ideal situation to go into the last year of contract without extending it,” Hasenhuttl said in last night’s fan forum. “Jannik is a fantastic player, he did a fantastic job in the last two years with us, he played a fantastic Euro with Denmark.
“Now we must have a look what happens, either he extends his contract or yeah when there is an offer and he has a chance to go to a bigger club he must always decide what to do.”
Along with Leicester, Vestergaard has attracted transfer attention from Tottenham Hotspur and West Ham United over the last couple of seasons.