The Dutchman was the subject of interest from both clubs last month, along with Manchester United and Chelsea.
But Southampton were reluctant to let him leave following the departure of centre-back Jose Fonte to West Ham.
Reports suggest Manchester City and Liverpool will battle it out again for the £50m-rated defender in the summer.
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Virgil van Dijk is under contract with Southampton until 2022
Premier League stats: Most tackles so far in 2016/17
Wed, February 8, 2017
Which Premier League player has made the most tackles so far in 2016/17?
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Which Premier League ace has made the most tackles so far this season?
But according to the Daily Star, Van Dijk has no plans to leave St Mary's after recently being appointed captain.
Saints sought assurances from the 25-year-old's commitment before handing him the armband and they are now convinced they can keep him for at least another season.
A source told the paper: “When Virgil took the captaincy it was on the understanding he would lead the team until the end of next season as a minimum.
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Southampton believe Virgil van Dijk will stay with them next season
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Jurgen Klopp may have to look elsewhere for defensive signings in the summer
“He has a long-term contract and is settled at the club.”
Having signed a bumper six-year contract last May, Van Dijk is under contract with Saints until 2022.
And with Southampton confident Van Dijk will stay in the transfer window, Pep Guardiola and Jurgen Klopp may be forced to look elsewhere to bolster their defence.
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