Jones was sacked by England earlier this month after a poor run of form.
"I'm very much looking forward to returning to Twickenham Stadium - a venue I have so many fond memories of," said Jones.
"I'm more motivated than ever to put on a show."
The World XV will be overseen by former All Blacks coach Hansen, against whom Jones and England achieved a 2019 World Cup semi-final victory.
"Going up against Steve rekindles a great rivalry which first started in our days competing in Super Rugby and progressed into the international arena," added Jones.
"It'll be a great challenge and one I'm sure we're both relishing."
Jones has been linked with a return to his native Australia and a job with the United States national team since leaving England, while various players - such as Owen Farrell and Anthony Watson
- have said they are disappointed to see him go.
The Rugby Football Union is yet to appoint a successor for Jones, who, despite a recent dip in results, has the best overall win percentage of any England men's coach in history.
Leicester head coach Steve Borthwick, who previously worked as Jones' forwards coach at England and Japan, is likely to fill the vacancy.
The Barbarians beat an All Blacks XV at Tottenham Hotspur Stadium in November before playing fixtures against Northampton, Bath and Harlequins in weekends left free by the demise of Wasps and Worcester.
Hansen still holds a director of rugby position with a Japanese club.