Fonte, 32, was strongly linked with a move to Old Trafford during the summer transfer window with the Portugal international revealing in September there had been interest from Jose Mourinho's side.
"Was I disappointed the Man United move didn't happen? Let's leave that for another time," he said earlier this season, according to Manchester Evening News. "Now I'm focused on Southampton."
Fonte has retained his place in the Saints' first team this season in the league alongside Virgil Van Dijk, but has been frustrated with Claude Puel's rotation policy that has seen him miss out on the club's Europa League campaign.
A number of reports suggest the Portugal international was offered a pay rise before the season started, but his club were unwilling to grant an extension to his contract that expires in 2018.
With uncertainty over his future at the club growing, Fonte has looked to clarify the ongoing situation with a message to his 258,000 followers on Instagram, writing: "Just to set the record straight I did not reject a new contract, in fact I have been informed by Southampton that they are not offering me a new contract."
Fonte was linked with becoming Mourinho's fifth summer signing at Old Trafford despite having already signed Eric Bailly from Villarreal during that transfer window.
Injuries to Bailly and Chris Smalling have seen Phil Jones and Marcos Rojo deployed at centre-half in recent weeks, but according to the Mirror, Mourinho is looking to bring in a "top level" defender during the winter window.