The Hammers are keen on the Burnley centre-back as they look to strengthen their defence in this transfer window.
Yet they face competition from Leicester for the 27-year-old – and the Telegraph report that the Foxes have submitted an improved for the defender.
Leicester have already had one bid rejected for Tarkowski but are said to be optimistic of being able to reach an agreement with the Clarets, with the new offer worth over £30m.
Burnley’s asking price is £50m but Leicester are keen to negotiate due to the financial impact the coronavirus has had on football.
In a further blow to the Hammers, Tarkowski is said to be interested in a move to Leicester.
Leicester’s willingness to negotiate and increased spending power could mean it is difficult to see the Hammers sealing a deal for the England international.
Speaking earlier this week, West Ham owner David Sullivan hinted the club were unwilling to meet Burnley’s valuation for Tarkowski.
Whilst he stopped short of naming the player directly, Sullivan said he would not pay £40m for a Premier League defender.
He said: “We’ve got a very unbalanced squad. We’ve now got the funds to buy a player or two, but unfortunately at the moment the players the manager wants we can’t get, and he’ll be spending 18 hours a day looking at types trying to find players.
“I’m waiting for targets for the manager and we’ve got two or three bids in, but unfortunately the benchmark is very high and it’s very hard to get those players.
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“We have got limited funds; if we had £400million to spend and someone said you spend 10 per cent of it – £40m – on a 28-year-old centre-back from a Premier League club who will remain nameless, at £40m you’d probably get the player out.
“For 10 per cent of your budget, it would probably make sense but to spend your entire budget on a 28-year-old centre-back…
“We’ve also got the fact we’ll probably get no gate money for the rest of the season and we’ve got to keep the club afloat and pay wages.”