Smith issued a statement requesting a trade on Tuesday, writing "the new front office regime doesn't value me here".
Set to make $9.735million in the final year of his rookie contract, Smith, who represents himself, had been holding out during training camp while working towards a new deal.
The 25-year-old, however, has become infuriated with negotiating with first-year general manager Ryan Poles and now wants out.
"The new front office regime doesn't value me here," Smith wrote in a statement posted by Ian Rapoport of NFL Network.
"They've refused to negotiate in good faith, every step of this journey has been 'take it or leave it'.
"The deal sent to me is one that would be bad for myself, and for the entire LB market if I signed it. I've been trying to get something done that's fair since April, but their focus has been on trying to take advantage of me.
"I wanted to be a Bear for my entire career, help this team bring a Super Bowl back to our city.
"However, they have left me no choice than to request a trade that allows me to play for an organisation that truly values what I bring to the table."
Selected eighth overall in the 2018 draft, Smith has been one of the league's top linebackers and was looking to be paid like one.
Smith was reportedly offended by Chicago's backloaded offer that included de-escalators that are not in the contract of any of the other 94 non-quarterbacks making at least $15m per year.
Smith ranked fifth in the NFL in total tackles last season with 163 to go with 12 tackles for loss to earn second-team All-Pro honours for the second straight year.
In 61 career games, he has 538 total tackles, 43 tackles for loss and five interceptions, making him one of just three defensive players with at least 300 tackles, 30 tackles for loss and five interceptions since 2018, along with Darius Leonard and Jamie Collins Sr.
Smith apologised to Chicago fans and implied he could possibly withdraw the trade request if team ownership can restore the fragmented relationship but said it is unlikely he will ever put on a Bears uniform again.
"I haven't had the chance to talk to the McCaskey family, and maybe they can salvage this," he wrote, "but as of right now I don't see a path back to the organisation I truly love."