The Sun on Sunday says Mourinho remains interested in signing Fonte after failing to land the defender during the summer transfer window.
However, club bosses have told him that he must trim one defender from this squad if he wants to sign the £10m-rated defender.
Fonte, 32, has made five league appearances for Southampton this season.
He starred for Portugal in their victorious Euro 2016 campaign and has a contract with the Saints until 2018.
Earlier this week, Fonte confirmed that he had not been offered a new contract by Southampton yet.
"There's nothing on the table, so I'm waiting," he was quoted as saying by the Sun.
"I'll see what happens. I'm just focusing on performing above all and we'll see what happens. We haven't agreed anything, so it's just talks.
"There were some talks but there hasn't been for a while. Like I said, I'm focusing on my job and we'll see what happens in the future.
"But I'm not concerned about that, just concerned about performing and doing well for my club."
Reports in the summer suggested Mourinho was keen to sign a defensive leader in the mould of former United captain Nemanja Vidic to marshal the back four.
The Portuguese coach signed one defender in the summer – Ivory Coast centre-back Bailly, 22, from Villarreal for £30m ($39m).
Aside from wanting more experience in the back four, Mourinho was also reported to be conscious of the fact that Bailly will be unavailable in January while he competes in the African Cup of Nations.
Full-backs Matteo Darmian and Marcos Rojo could be most at risk if Mourinho decides to pursue a deal for Fonte in the January transfer window.