Voetbal International says the two Premier League clubs are in talks with Celtic about tempting the Dutchman south.
Van Dijk is tied to Celtic until 2017, but is tempted by a move south.
The centre-half is likely to be on the move after Celtic were thumped by Champions League opponents Legia Warsaw last night.
Both Swansea and Sunderland are in the market for a new centre-half signing this summer.
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