The 27-year-old has starred for the Clarets since signing from Brentford back in 2016. He was in particularly impressive form last season, registering more clearances than any other Premier League player.
The Hammers had an initial offer - believed to be in the region of £23m - rejected last week. According to Sky Sports, a second bid of £30m has also proved unsuccessful. Burnley are understood to be holding out for £50m as Tarkowski is one of their star players.
Leicester are another side that have been interested in Tarkowski in the past, however a source has told 90min that the Foxes have been put off by this hefty asking price. Brendan Rodgers has instead earmarked Saint-Étienne prospect Wesley Fofana as his number one transfer target. Talks are already underway with the Ligue 1 side willing to let the 19-year-old depart for only £20m.
With Leicester out of the race, West Ham's attempts to sign Tarkowski will be unchallenged. David Moyes is a big fan of the defender and is keen to make some new signings following the controversial sale of academy graduate Grady Diangana to Premier League rivals West Brom.
The sale attracted impassioned criticism from Hammers players and fans alike, with Mark Noble and Jack Wilshere among those hitting out at the club. Signing Tarkowski could help to quell this unrest, although there is some way to go before a deal is struck.
With so much turmoil off the pitch, Moyes will be desperate for his side to get off to a winning start in their first game of the season against Newcastle on Saturday. The Hammers follow this up with a London derby against Charlton Athletic in the EFL Cup.
Source : 90min