Posh accepted a £3million offer from their Championship rivals earlier this week and the Scotland international is now discussing a move to the Walkers Stadium.
Mackail-Smith has already reportedly rejected moves to Premier League duo Norwich City and Queens Park Rangers, while West Ham United have also shown an interest.
Last season's League One champions Brighton & Hove Albion also made a late bid for the 27-year-old, but it appears he is on his way to Leicester.
"I would expect him to sign for Leicester on Friday," said Peterborough chairman Darragh MacAnthony.
"Brighton came to the table with a strong offer but it wasn't anywhere near as strong as Leicester.
"West Ham made a very serious enquiry. I believe they're still very interested. I presume if Sven gets on with Craig he'll be a Leicester City player on Friday."