This season, Gibson has averaged 11.6 points and 7.0 rebounds over 55 games.
McDermott offers the Thunder another outside scoring option - he's averaging 10.2 points a game, while shooting 37.3% from the three-point arc.
He represents an upgrade on Morrow, who has previously filled the same role with OKC, averaging 5.8 points and shooting 29.4% from long distance.
For the Bulls, the most attractive piece of the swap would be back-up point guard Payne.
Chicago have struggled at the point, with veteran Rajon Rondo unable to spark their starting line-up, so they'll be keen to give Payne a chance in their rotation.
Meanwhile, the trade news was not so good for Australian veteran Andrew Bogut, who was moved from Dallas Mavericks to Philadelphia 76ers, in exchange for younger, more athletic Nerlens Noel.
Bogut, who won an NBA ring with Golden State Warriors in 2015, will likely be bought out by the re-building Sixers, making him available to other teams that value his experience.
Among those most interested in Bogut are defending champions Cleveland Cavaliers.