Versteeg is the last of the Hawks' seven restricted free agents believed to have received qualifying offers after the June 29 deadline. The Blackhawks re-signed the other six on Monday.
The National Hockey League Players Association filed a grievance Monday over the Hawks' handling of the qualifying offers. If the NHL rejects the grievance, the matter will go to an arbitrator. At that point, it would be possible for Versteeg to be declared an unrestricted free agent, and the Hawks could lose the talented, young forward on the open market.
But Versteeg's agent David Kaye said he doesn't want it to get to that point.
"We've been negotiating with Chicago fairly regularly here, and I expect hopefully something will get done in the next few days," Kaye said Tuesday.
When asked if it's possible a deal will be reached Tuesday, Kaye said: "You never know. You know how negotiations are."
Kaye said he has no idea what the timeline would be for the arbitration process. He also preferred not to discuss the Blackhawks' apparent paperwork error that led to this point.
"I don't want to get into it right now," he said. "We're in the middle of negotiations. We've been negotiating on an ongoing basis."
But when asked if Versteeg, who led all rookies with 31 assists and was a finalist for the Calder Trophy, wants to stay in Chicago, Kaye said: "Most definitely."
"[Versteeg is] good," Kaye added. "He's excited, as any young guy would be; excited, nervous."