Van Dijk is wanted by Chelsea, Liverpool and Arsenal and all three will pay £60m to get him, The Sun claim.
They claim the Dutch defender has had a series of rows with Saints top brass over being allowed to leave.
He’s had face-to-face meetings to get his point across, but the south coast club are holding firm.
Van Dijk has proven his class as a top Premier League performer since joining Southampton from Celtic in 2015.
Saints will turn a huge profit on the £11.5m they paid to sign him but they are standing firm and refusing to sell, The Sun report adds.
Van Dijk signed a new five-year contract last summer, so they are under no pressure to cash-in.
John Collins, who knows the defender from his time at Celtic, thinks his agent made a mistake in the negotiations.
He said earlier this month: “Virgil van Dijk, I know Virgil very well.
He’s a good boy but I don’t know why he signed this six-year extension last year to be perfectly honest.
“When you sign an extension six months earlier, unless your agent has got a buyout clause put in it and a buying club makes it, then his agent’s not done a good job for him.
“It looks like Southampton are going to dig their heels in and it looks like Liverpool are going to dig their heels in.”