The former Germany midfielder, who was made to train on his own earlier this season, was a late substitute in Wednesday's EFL Cup win over West Ham - his first appearance of the season.
The 32-year-old has spoken to Major League Soccer side Chicago Fire and had contact with other clubs in the USA.
"Of course he can [stay]," said Mourinho on Friday.
"If you have a contract, you are the powerful one that can decide your future. In January he is the powerful one - and with better feelings than before.
"Everybody was happy because when he was not training he fought against that and managed to do it without injury.
"He had been working for months and I thought it could be a good opportunity for him to play and help us."
Mourinho's side, who are sixth in the Premier League table, travel to play Everton - who are a point and a place behind them - on Sunday (16:00 GMT).