And Danijel says the club should accept after paying 20m to sign his brother from Stoke City in the summer of 2017.
Reports have stated the offer - which, it is understood, would mean Arnautovic earns approximately £40m over his four-year contract, a vast upgrade on his current wage of £90,000 a week - is from reigning champions Shanghai SIPG, although their permitted allocation of three overseas players is filled by former Chelsea midfielder Oscar and his fellow Brazilians Hulk and Elkeson.
It is understood the club, as yet unnamed, will make an improved bid and offer lucrative terms in excess of £200,000-a-week for the Austrian global.
Pellegrini added that the two-time Premier League winner's previous experience sets him in good stead for his return to football with West Ham.
The statement, issued to the radio station Talksport, went on: "He knows the West Ham fans love him and he has a special connection with them, but things move on, this is the world of sport, and he hopes they can understand". He will give 100 per cent for West Ham.More news: Dallas Cowboys Owner Jerry Jones Splashes Out on Superyacht
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Pellegrini, who substituted Arnautovic against his will after only 20 minutes in Saturday's FA Cup win over Birmingham, had earlier shrugged off the speculation. Marko in this moment is a player of our team, players of his quality and other players like Declan Rice, Felipe Anderson, other clubs will want them but that is one thing.
'I think Marko and all the other players involved in rumours have their heads here and want to continue playing the way they are doing so far.
Marko is professional and is preparing himself for the Arsenal game.
"If it was £50m between Wilson or Arnautovic I would go with Marko", he told The Debate on Sky Sports.
While some Hammers seem adamant that Arnautovic won't leave during the January transfer window, a plethora of fans have fired into the Austrian amidst the transfers rumours, and here's a flavour of what some of the fans have said about these recent goings-on and many have likened this situation to that of the Dimitri Payet saga.
"I want to keep him, of course, but you never know what might happen in the future".