The Bears this week gave their prized lineman a franchise tag, which gives the two sides until July 15 to work out a long-term deal.
"Henry has been a good player for us and we are using the franchise tag so his development continues as a Bear," general manager Phil Emery said on the Bears' website. "We will continue our conversations with Henry and work toward a multiyear agreement."
The Chicago Tribune said Saturday it appeared there was a long way to go in the discussions.
Melton is reportedly seeing a five-year deal, likely worth $18.6 million, which is about the minimum for a franchise player over two years. The team, which is juggling salary cap restrictions, may have to restructure the contracts of some of their other veteran players.
Melton, a fourth-round draft selection four years ago, was tied for third in the league among defensive tackles with six sacks.