Burnley defender Michael Keane is resisting signing a new deal at Turf Moor.
The People says Leicester tabled a bid of £10m which could have risen to £11.5m with add-ons, then followed that up with a second offer which started at £12m but could have ended up being worth £15m.
Burnley boss Sean Dyche does not want to lose one of his best players from a side that has won promotion back to the Premier League.
But Keane is eager for the move to happen.