But the newspaper claims the Clarets are in no rush to sell, with the 26-year-old's contract running until 2022.
And it is suggested that Wolves or Leicester would have to pay a 'huge' increase on the £3 million fee Burnley shelled out to Brentford three-and-a-half years ago, if they are to tempt them into a sale.
But how much do those of a Clarets persuasion want for the England international - and do they even want him to stay?
This is what's being said on Twitter...
Double it !— The Curtain loft (@paulcurtainloft) June 14, 2019
Keano went for 30m 2 years ago. In today's market Tarky is worth at least 50m— Bradley. (@BJ_Sutcliffe) June 14, 2019
John Stones went for 50m. Everton got good value for Keane. That’s the benchmark now. 50 and no less.— Chris Lambert (@chrislambo88) June 14, 2019
If we was to get the trigger of 50 mill i would drive him my self crazy crazy money for a centre half.— scott mitchell (@scottburnley) June 14, 2019
Has Gary Cahill moved on yet? Sell Tarks buy him if not?— Andrew 'Skinny' Wray (@skinnywray) June 14, 2019
40 mill yes please,whoever mee plays with turns into an England international Gibson more than fits the mould thank you :)— welsh claret (@nathanwa11) June 13, 2019
Take the £40m, install Gibson. Look for the next Mee, Keane, Tarks. That’s the way we roll. The tactical unit is far stronger than the individual, see Shackell, Keane after leaving. Gibson more than adequate. I’m easy either way.— Oliver Moorhouse (@longsideliveson) June 13, 2019
I've got the engine running where's he live?— martin spiers (@claretbod) June 13, 2019
You can dream, Tarky is staying at Burnley— All Northern Soul (@RoyAsten) June 14, 2019
Tarkowski is being touted as a potential Harry Maguire replacement at Leicester, while his position is one in which Wolves are expected to strengthen this summer, with Europa League football added to the club's schedule next season.
Can Burnley keep hold of the reported Wolves and Leicester target?
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