The Italian side are said to have offered cash plus midfielder Felipe Melo in part-exchange for Portuguese international Meireles.
Liverpool have apparently responded by declining the offer but indicating that they would be interested in a straight cash deal of £13.5m-£15m. He arrived from Porto last summer for £10.7m.
While Meireles enjoyed a successful debut Premier League campaign, scoring five goals in 33 appearances, Liverpool are said to be open to losing him due to a hefty midfield roster, including newly-acquired Jordan Henderson.
Meanwhile, Melo has expressed his frustration with his part in proceedings. “Let us say that Juve are not respecting me, or at least that’s the impression I get,” The Mirror quotes him as saying.
“I’m not just anyone. I’m not a bargaining chip to be used to reach anybody.”