However, it is believed that the club are ready to change their stance as they are aware his desire to play in England outweighs their ability to retain his services.
Southampton chief Ronald Koeman has made clear his desire to recruit another top-class centre-half and has not hidden his admiration for his fellow Dutchman.
Despite being early days, Van Dijk has been a star performer for the Bhoys so far this season. Having made two league appearances to date, he has ensured two clean sheets for his side, while showing fantastic composure in defence.
The 52-year-old head coach will look to the club's previous successes when recruiting from Celtic, with 'keeper Fraser Forster and midfield enforcer Victor Wanyama fine examples of player who have made the transition from Scotland to the English top flight.
Southampton will have to wait until after Celtic's Champions League play-off with Malmo before signing the highly-rated defender, but the club are confident that the transfer will go through before the window shuts on September 1 - regardless of the outcome of the tie.