The Foxes are long-term admirers of the defender and have made an offer that includes add-ons, the PA news agency understands.
They had two bids of around £30million rejected in January but have returned as Brendan Rodgers looks to improve his defence.
Burnley are understood to want £35million up front, with £10million of add-ons, and Leicester’s opening bid is likely to fall short of the Clarets’ asking price.
Tarkowski would be keen on a move to the King Power Stadium as he looks to improve his England chances.
He is yet to add to the two caps he earned in 2018 and is understood to have been frustrated at being overlooked for the Three Lions.
West Ham has two bids rejected earlier this month but Leicester are now in pole position.
Tarkowski moved to Burnley from Brentford in 2016 and has played 143 times, scoring five goals.
Leicester boss Brendan Rodgers has been vocal about his desire to sign another defender and had been considering St Etienne’s Wesley Fofana.
But Tarkowski has been on his list and the Foxes can offer him Europa League football this season.
