The 20-year-old has impressed in Serie A last season notching up seven goals and two assists in 17 starts.
The Uruguay international joined Bologna from Penarol in the summer of 2010 and has a contract with the club until 2015.
He is being tracked by a number of clubs believed to include Inter Milan, Roma and Barcelona, (according to the Sun.)
But the Reds hope they can use the good friendship with between Ramirez and current Anfield favourite Luis Suarez as a deal breaker much in the same way the hit-man helped lure Sebastian Coates to Anfield.
Joining Kenny Dalglish’s Anfield rebuild would certainly be interest the winger who is reportedly keen to test himself at the highest level available.
Bologna had a poor season last term, finishing in the bottom half of the table (16th) and not much has changed so far in this campaign, currently sitting 19th with no points thus far.