The 35-year-old, who is recovering from knee ligament damage, scored 28 goals in 46 appearances for the club last season but was released when his previous deal expired at the end of June.
However, the striker has agreed to return to Old Trafford and manager Jose Mourinho expects him to be a hugely important player when he returns to full fitness, which is expected to be in January.
"We are delighted Zlatan is on the road to recovery and we are equally delighted to have his ambition and experience back with us," Mourinho told United's official website.
