The Scotsman spent last season at loan with Aston Villa.
Payet was incredibly difficult to replace but Arnautovic can give West Ham United what they never got from the Frenchman.
West Ham knocked Chelsea out of the EFL Cup in an end-to-end London derby that finished 2-1.
Selling Dimitri Payet for £38 million is good business for West Ham, but they should not do it.
West Ham's misfortune will be Tottenham's gain in a fortnight.
The Hammers will play their home games in the Olympic Stadium from the 2016-17 campaign.
Bilic's first game in charge as Hammers boss on Saturday gave the Croatian plenty of food for thought.
The 26-year old has been named as the Hammers captain for tonight's Europa League qualifier.
The Magpies are still to make their first summer signing as pre-season gets under way.
The Hammers are preparing for their Europa League first round qualifier against FC Lusitans on Thursday.