The Frenchman had been lined up for a reunion with Claude Puel at St. Mary's.
Former Nice man Jeremy Pied
The 27-year-old, who is a free agent after leaving Nice, is now said to be weighing up alternative offers.
New Saints boss Claude Puel gave Pied his debut at Lyon, and took him to Cote d'Azur two years later in 2012. However, a reunion at St. Mary's now seems unlikely, after club officials reportedly offered terms below his expectations.
New Saints boss Claude Puel
Southampton already have Euro-2016-winning right-back Cedric Soares on their books, with Cuco Martina acting as back-up during the 2015-16 campaign.
Portugal's Nani, Cedric and Joao Moutinho celebrate after winning Euro 2016
Pied, though, can also operate as a left-back or winger on either flank, the latter of which could come in handy given Sadio Mane's recent departure to Liverpool.
Norwich City winger Nathan Redmond arrived at St. Mary's last month, while another Canaries wide man, Robbie Brady, has also been linked (the Mirror).
Republic of Ireland's Robbie Brady scores against Italy
Pied appeared 33 times for Nice last term and is a former France Under-21 international.
The Lyon product, who won the Coupe de France with in 2012, has also enjoyed loan spells at Metz and Guingamp