The person spoke Thursday on condition of anonymity because the deal had not been announced. FanRag sports first reported the deal.
Jennings made a career-high 64 appearances last year in his second season with the White Sox, going 4-3 with a 2.08 ERA. His 46 strikeouts matched a personal best set in 2015.
The White Sox finished fourth in the AL Central at 78-84 last year -- their fourth straight losing season. They promoted Rick Renteria from bench coach to replace Robin Ventura as manager and traded ace Chris Sale to Boston and outfielder Adam Eaton to Washington for hefty packages of young players.