The 24-year-old has been on the Reds' radar for the past 12 months and had been linked with a summer switch to Anfield.
However, according to reports in Italy, it is Sassuolo who are currently frontrunners to land the £8m-rated ace.
Saponara played 33 times in Serie A for Empoli last term, scoring five goals, and the club do not want to sell him.
However, it has been suggested that will not stop Sassuolo from firming up their interest with a bid, and they think any offer over £7m should be enough to make Empoli sell.