Ibrahimovic had been widely expected to join the English club after informing Paris St Germain that he was leaving after four years in the French capital.
The much-travelled striker will link up with Mourinho, his former manager at Inter Milan, who has just taken over at Old Trafford.
United, who have not paid a fee to sign the out-of-contract Ibrahimovic, will be the eighth club of a career which began with his hometown team Malmo in 1999 and has included spells at Ajax, Juventus, Inter Milan, Barcelona and AC Milan.
He has scored 392 goals in 677 games and won a trophy in each season of his career since 2001.
Beyond the silverware, however, it is the box-office appeal of the Swede which has appealed to United, whose supporters have bemoaned the lack of star-quality in their teams since the departure of Alex Ferguson.
United were widely criticised for playing boring football last season under now departed manager Louis van Gaal. Boring is not a word which could be used to describe the flamboyant Ibrahimovic, who scored 50 goals in 51 matches for PSG last season.
United have not yet confirmed the deal but British media reported that he will undergo a medical at Old Trafford on Friday.
Ibrahimovic announced his retirement from international football last week after Sweden’s early exit from Euro 2016.
He will link up with his new team mates next month and could feature on the club’s tour of Asia, for which they are due to depart on July 19.