The Foxes are believed to want both top strikers this winter.
The club are already in talks to finalise a move for Croatian international Andrej Kramaric. The £6 million goal machine has been discussing a deal with Leicester this week after reportedly rejecting an approach from Chelsea earlier this month.
Sky Sports believe that Leicester will bid for Burnley's striker this month.
The England under-21 international is a pivotal player for The Clarets, scoring five goals for the club this term as they also attempt to fight off relegation.
Leicester would have to pay in excess of £7.5 million for the striker, as Burnley reportedly rejected such a bid in December.
The likelihood is however that Ings will not leave for Leicester. He made it clear at the end of last month that he was not keen on an exit from the club and Burnley are unlikely to sell their best striker to a direct relegation rival.