New Toffees boss Silva has the backing of football director Marcel Brands to try and tempt the Hammers who value the striker at £50million.
That is more than double the figure West Ham paid Stoke for Arnautovic last summer, but it would not faze Everton who are backed by wealthy owner Farhad Moshiri.
But they would have to outmuscle Manchester United who remain favourites to land the 29-year-old this summer.
United are keen to offload some high earners before bringing in new recruits, which could open the door for other clubs to bid.
And Everton are set to go in early for the Austrian whose 11 Premier League goals saved West Ham from the drop last term.
Silva believes Arnautovic’s pace and power can ignite an Everton squad who, last season, created fewer chances and had less shots on target than any other top-flight club.
United boss Jose Mourinho wants Arnautovic and the West Ham talisman would find it impossible to turn down the chance to work with Mourinho again.
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The pair spent a season together at Inter Milan in 2009-10, which was marred by injuries to Arnautovic.
And Mourinho sees the Hammers’ hitman as the perfect foil for Romelu Lukaku.
The Portuguese coach was impressed by Arnautovic bullying United’s defence — in particular Chris Smalling — during a 0-0 draw at the London Stadium in May.
He believes he has the pace and power to terrorise Premier League defences in a Red Devils shirt, either playing off Belgian star Lukaku, leading the line, or playing on either flank and cutting inside.
Mourinho has told United’s board he would be getting FOUR forwards for the price of one, with Arnautovic also capable of playing as a No 10 behind Lukaku, 25.
The stumbling block, for both United and Everton, will be West Ham’s fear of more protests from fans if they sell their top player this summer.
They have seen Stoke go down after they offloaded Arnautovic for £24m last July.
Although he has four years left on his current Hammers deal, if United were to come calling, Arnautovic would jump at the chance to join one of the biggest clubs in the world.
He knows it would be a once-in-a-lifetime move and West Ham’s new boss Manuel Pellegrini may not fancy kicking off the season with an unsettled striker — if the move was blocked.
West Ham’s bigwigs might feel doubling their money for a player who turns 30 next season, is not bad business, with Chilean Pellegrini looking to bring in his own men.
The east London club are also lining up Paris Saint- Germain’s Javier Pastore, 28, to bolster their attacking options should Arnautovic leave this summer.
Talks with the French champions are ongoing over a deal worth up to £17.5m for the 6ft 2in Argentine playmaker.
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