McCown featured regularly for the Bears this season due to a number of injuries to starting quarterback Jay Cutler, leading the side to a 3-2 record with figures of 1,829 passing yards, 13 touchdowns and just one interception.
The 34-year-old is due to become a free agent this off-season, but he has no plans to call it a day, according to ESPN.
The Bears are reportedly keen to extend McCown's contract, while the Minnesota Vikings and Tennessee Titans are also on the lookout for available quarterbacks.
This year's Draft does not include as many quarterbacks as expected, meaning that veterans such as McCown and Michael Vick, who will also become a free agent during the off-season, could be in high demand.
McCown has been with the Bears since 2011.