Ramirez joined the Saints back in 2012 and made a positive start to life at St. Mary's, but he quickly fell out of favour and has not been given an opportunity under Ronald Koeman.
Ramirez has not played for Southampton since August 2014, with his most recent football coming on a loan spell with Hull City last season.
That temporary move did not work out as planned, with the Tigers going down and Ramirez failing to make too much of an impact, but Lazio are still reportedly keen on signing the Uruguayan.
The Serie A club are looking to add creativity to their ranks, and feel Ramirez could offer that - but they face competition from two of Ramirez's former clubs.
Both Bologna and Penarol are keen on a reunion with the 24-year-old and will apparently offer him an escape route when the January transfer window opens.
Southampton are unlikely to stand in the way of any deal, with Ronald Koeman reportedly having told Ramirez that he has no future at the club.