Crystal Palace have long been credited with interest in the West Ham United winger.
The newspaper claims Palace are considering selling Andros Townsend back to Newcastle United, while Antonio will be allowed to leave West Ham.
And it adds that Townsend's move would also spark Matt Ritchie's transfer, from Newcastle to Stoke City.
Crystal Palace have long been credited with interest in Antonio, although the Daily Mirror reported in May that the 28-year-old would be difficult to prise away from West Ham.
That no longer appears to be the case, however, after the Hammers signed Ukrainian winger Andriy Yarmolenko for £22 million earlier this week.
According to Transfermarkt, Antonio has scored or made 32 goals in 90 games since he arrived at West Ham from Nottingham Forest three years ago.
And the south Londoner was called up to England's senior squad as recently as March 2017.
Crystal Palace have also been linked with a move for West Ham centre-back Declan Rice, as reported by the Sun last month.