Ibrahimovic’s Old Trafford career appeared to be over after his first season in England was cut short by a cruciate-ligament injury in April and United decided not to extend the 35-year-old striker’s contract.
But Mourinho said: ‘Last year was not enough for him. He wants more football at the highest level, so we are discussing the possibility of him staying for the second part of the season.’
Despite spending £75million on Romelu Lukaku, United believe there is a place in the squad for Ibrahimovic, who scored 28 goals last season.
He continued his rehab at Carrington until July 15, two weeks beyond the end of his contract, and United were impressed by his rate of recovery.
The striker, who was the league’s highest paid player on £367,000 a week last season, believes he has unfinished business at United and has been encouraged by the club’s goodwill.
AC Milan also want Ibrahimovic, who has rejected offers to wind down his career in America and China.
Mourinho has won the title in his second season at every club he has coached and, ahead of Sunday’s opener against West Ham, he said: ‘A manager in the second season knows the club and players better.
‘We are just in a position to do quality work better than in the first season. It’s not just hope, it’s a conviction that the quality of the team will be better.
‘We have to put ourselves in a position that allows us to fight for the title. If not, finish in the top four.’