The 23-year-old had been linked with Premier League sides Sunderland and Southampton during the summer, with the Hoops rebuffing all offers for the Dutchman.
It is thought that Gunners boss Arsene Wenger received a glowing report when he sent his scouts to watch the centre-back in Celtic's Europa League clash with Dinamo Zagreb a fortnight ago.
The Hoops have suggested that they are in no hurry to sell any of their key players, after goalkeeper Fraser Forster joined Southampton in August.
However, Arsenal are prepared to test the Scottish side's resolve by making an £8m bid for Van Dijk, according to The Scotsman.
Van Dijk joined Celtic from Groningen in the summer of 2013.