The 26-year-old has been heavily linked with Liverpool although the Merseyside club officially ended their interest after Southampton reported them to the Premier League for an illegal approach last month.
Saints do not want to sell Van Dijk but last week manager Mauricio Pellegrino said the centre-half was training alone because he is "not 100 per cent" mentally.
Van Dijk, who is valued at 60 million pounds, missed Saturday's friendly at Brentford and has now been omitted from Southampton's 25-man squad for France, where they will play Saint Etienne on Saturday.
British media reported that Liverpool remain interested in the defender, who last year signed a six-year contract at Southampton, but are wary of angering their Premier League rivals with another approach.
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Liverpool already have four former Southampton players in their team, having spent 91 million pounds ($118.64 million) on Adam Lallana, Dejan Lovren, Nathaniel Clyne and Sadio Mane since 2014. A fifth player, Rickie Lambert, was signed and then sold on.
Liverpool's situation is further complicated by their own stance on midfielder Philippe Coutinho.
Any further move for Van Dijk, unless sanctioned by Southampton, would risk accusations of hypocrisy, given Liverpool's reluctance to sell Coutinho despite strong interest from Barcelona.
(Reporting by Neil Robinson; editing by Mark Heinrich)
