The Dutch tactician was in charge at St Mary's when the club paid a fee in the region of £13million to land the 25-year-old from Celtic last summer, and he is under contract until 2022 after signing an extension in May.
Van Dijk's form under Koeman last term, and also current boss Claude Puel this season, has resulted in a growing number of suitors thought to be preparing massive bids to take him as early as the January transfer window.
Manchester United and Liverpool were initially linked while reports over the weekend suggested that Manchester City boss Pep Guardiola wants to take Van Dijk as a long-term replacement for injury-ravaged skipper Vincent Kompany.
But following Van Dijk's display against Koeman's Everton side on Sunday, when he helped keep a clean sheet in a 1-0 win over the Toffees, reports are now suggesting the 53-year-old would like to be reunited with his fellow compatriot on Merseyside.
Van Dijk has already stressed that he is focused on playing well for Southampton while Puel will no doubt do all he can to resist any big-money offers in order to keep one of his key men on the south coast.