Southampton manager Ronald Koeman
The BBC report the Goodison Park outfit will pay about £5m compensation for the 53-year-old Dutchman, who leaves Saints after two years in charge.
The Blues have been without a manager since they sacked Roberto Martinez just before the end of the 2015-16 season.
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Who will net Neto?
But with Koeman's appointment not set to be confirmed until Friday at the earliest, it could give Slaven Bilic's Hammers an advantage in a bid to land Brazilian goalkeeper Norberto Neto from Italian giants Juventus.
West Ham manager Slaven Bilic
Advantage West Ham
With Koeman still not in place, Bilic has a real window of opportunity to steal a march on Everton for the talented 26-year-old.
West Ham's impending move to the 60,000 seater Olympic Stadium this summer could also sway any decision Neto has to make as could the fact the East Londoners are in the Europa League next season while the Toffees are not.