The 22-year-old has become a key man for the Foxes this season, making 41 appearances and scoring seven times.
According to the Mirror, Barnes' performances have caught the eye of Jurgen Klopp.
The Liverpool boss is looking to add to his attacking options this summer and Barnes has the ability to play across a number of positions.
His versatility and energy is said to appeal to Klopp.
Wolves' Adama Traore is also said to be on Liverpool's radar, but would command a huge fee.
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Leicester, similarly, would be reluctant to lose Barnes and have no need to sell.
The Foxes could secure Champions League qualification by beating Manchester United on Sunday - or bettering Chelsea's result - and look to have a bright future under Brendan Rodgers.
Rodgers has seen James Maddison commit his long-term future to the club and Barnes is also seen as key to Leicester's plans moving forward.
Liverpool are reportedly eyeing a shock move for Leicester City forward Harvey Barnes.