The report states Sunderland are 'understood to want up to £20 million' to part with Kone, for whom West Ham 'believe a deal could be agreed at just over half that' amount.
Sunderland defender Lamine Kone
A member of the Sunderland side which finished bottom of the Premier League last season, the 28-year-old was controversially identified by teammate Darron Gibson earlier this week as one of several Mackems players who no longer wish to be at the club.
Nevertheless, Kone impressed during his first half-season at Stadium of Light, helping Sam Allardyce's then-side to avoid relegation at the expense of their rivals Newcastle United.
And several of West Ham's Premier League rivals are said to have had an interest in the Ivorian, with Everton, Burnley and West Bromwich Albion each named by The Mirror.
West Ham currently have four recognised centre-backs on their books, although two of those - Winston Reid and James Collins - have been linked with moves away from the London Stadium in recent weeks.
West Ham defenders James Collins (left) and Winston Reid
Manager Slaven Bilic has already added Joe Hart, Pablo Zabaleta, Marko Arnautovic and Javier Hernandez to his Hammers squad this summer, at a reported cost of more than £40 million.
Should West Ham sign Sunderland's Kone?