The Uruguay international was a club-record signing for Saints in 2012 but started just 28 matches in four years at St Mary's.
Ramirez was frozen out by current boss Ronald Koeman and spent last term on loan with Hull before ending the current season at Middlesbrough.
Newly-promoted Boro are understood to be keen on signing the 25-year-old permanently after he scored seven goals in 15 Championship starts.
Veteran goalkeeper Kelvin Davis is also out of contract but is in talks over a new role behind the scenes at the club.
The 39-year-old joined Saints when they were a League One club in 2006 and had a testimonial last week.
Executive director of football Les Reed said: "Kelvin has been instrumental in the club's success in recent years, playing a key part in what has been a special time for everyone involved
"He has set the perfect example since he arrived at the club 10 years ago
"Kelvin's tremendous passion and commitment to the club are two of the reasons why we are currently in conversations with him about a new role moving forward, in which he can continue to help the club's progression."
Young goalkeeper Will Britt has also left Saints after his contract expired.
Source : PA