This was rejected by Leicester, however, with the East Midlands side looking for closer to £10 million.
Reddy reports that there has been “no progress” in talks between the two clubs, and Liverpool would need to Foxes to “soften their stance” in order to secure a deal.
Chilwell has just one year left on his current contract at the King Power Stadium, and is said to favour a move to Merseyside.
This puts Liverpool in the favourable position of luring Chilwell to Anfield on a free transfer next summer, with a compensation fee to be decided by tribunal.
In the meantime, Klopp is believed to be content to put his faith in Australian left-back Brad Smith this season.
The 22-year-old has handed a new contract by the German in November, after he impressed Klopp at Melwood following a summer in limbo.
Smith would play backup to Alberto Moreno, despite concerns about the Spaniard’s performances in the second half of 2015/16.
Liverpool have also reportedly pulled out of negotiations involved Udinese midfielder Piotr Zielinski, leaving Klopp to pursue alternative targets.
Meanwhile, Klopp is in the market for an experienced centre-back to cover injuries to Mamadou Sakho and Joe Gomez, while veteran goalkeeper Alex Manninger is currently on trial, looking to earn a short-term contract.