Premier League champions are expected to return with a new offer for Clarets centre back Michael Keane.
Burnley rebuffed the Foxes’ initial gambit, thought to have been around £12m - less than the £15m reported in some quarters - on Tuesday.
The offer for the 23-year-old former Manchester United youngster would have smashed the £7m figure received from Southampton for Jay Rodriguez in 2012.
But I understand Leicester have not been put off by seeing their opening bid turned down, and are prepared to up the ante to land one of the best young centre backs in the country.
The club have no need, or desire, to sell one of their crown jewels after a hugely impressive season, playing all bar two of Burnley’s league games as they clinched the Championship title.
Sean Dyche will receive the biggest transfer budget in the club’s history – with Burnley set for a massive windfall with the new Premier League television deal kicking in this season.
However, it remains to be seen whether he feels he could use the extra funds to keep the Clarets in the top flight, having also spent a reported £3m to land James Tarkowski – who filled in for Keane in February – from Brentford in the January transfer window.
Keane – who spent half a season on loan at Leicester in 2012/13 – is also believed to want the chance to speak to City about a potential switch, given the carrot of Champions League football, and the challenge of trying and oust Wes Morgan or Robert Huth, the miserly centre back pairing who were the bedrock of Leicester’s Premier League title success.
Claudio Ranieri has already signed a centre back this week, landing Real Madrid academy product Luis Hernandez on a free transfer, after the 27-year-old left Sporting Gijon at the end of his contract.
The Foxes are not the only club who have cast envious glances at Keane, with Everton, while Roberto Martinez was in charge, having previously earmarked him as a replacement for John Stones, a long-term target for Chelsea and Jose Mourinho.
The Toffees were understood to be prepared to pay an initial £6m, rising to around £10m with incentives.
Stoke City were also linked with a £7m switch in the January transfer window – the figure consisting of an initial £5m plus add ons – although a reported offer failed to materialise.
Keane has also been linked with Bournemouth, where former Clarets boss Eddie Howe is an admirer, and Burnley’s opening day opponents Swansea City, as well as Norwich City and Aston Villa before their relegation to the Championship.
Keane left Old Trafford for Turf Moor two years ago, initially on loan, before signing a permanent deal for a fee in the region of £2m in January 2015.
The former England Under 21 international has two years remaining on his contract with Burnley.