Brendan Rodgers wants fresh faces adding to his Leicester squad before the transfer window closes next week and is an admirer of Harrison.
The Foxes have now backed their manager by sending Leeds a bid of £20m for the winger, as they look to do a deal.
Leeds have yet to decide whether to accept the offer or to look to lock down Harrison to a new contract to keep hold of him.
The winger is now into the last 18 months of his contract at Elland Road and selling him now might give Leeds the maximum value.
However, it is unclear whether Harrison wants to join Leicester.
If Harrison pushes for a move to Leicester, Leeds could then look to cash in and sell him to the Foxes.
The winger started for Leeds away at Accrington Stanley in their FA Cup fourth round tie on Saturday afternoon.
