Fonte, 33, joined in 2010 and has turned down an extension to his contract which ends in 2018, according to club director of football Les Reed.
"It is important for me to keep stability in the defence with players," said Southampton boss Claude Puel. "We don't know the future for Jose.
"It is a difficult situation."
Puel did not select the Portugal international for Saturday's FA Cup draw at Norwich.
The former Lyon boss told the media that speculation over Fonte's future had been a distraction before the meeting with Liverpool at St Mary's, as Southampton bid for a first domestic final since 2003.
"I respect Jose and this is not easy for him, the squad or the staff," said Puel. "He is a good professional, but we have to respect this situation."
Fonte joined Saints from Crystal Palace in January 2010 and has made 288 appearances.
He was linked with a move to Manchester United in the summer after helping his country win Euro 2016.
Southampton defender Jose Fonte will not play in Wednesday's EFL Cup semi-final against Liverpool after making a transfer request.