Cheveldayoff has started the process, saying he had discussions with some of the free agents and negotiations with others. He would not specify who.
"The semantics of the two (discussions and negotiations) are important," he said, "but there are lots of different things, so to get into specifics about one or the other, sometimes it's just better off leaving them lumped together."
Stastny is a priority for the Jets, but it's unclear how the salary cap of $79.5 million for the 2018-19 season will affect their ability to get him under contract, especially with Hellebuyck, Trouba and Morrissey in line for significant raises.
"It really comes down to the cap," Cheveldayoff said. "People who get to UFA have earned that right [to look elsewhere], but [Paul] is and was very good for us, so if there is a chance we could keep him, we're going to certainly try to sharpen our pencil. If we can't, we're a better organization for having him for the period of time that we did. … We hope there is a chance we can find a way."
Signing Stastny and keeping all the RFAs might be impossible, which makes Cheveldayoff a GM to watch at the 2018 NHL Draft, which begins Friday in Dallas (7:30 p.m. ET; NBCSN, SN, TVAS). Trades are a possibility, because he said he doesn't expect to wade into the free agent market beyond Stastny.
"Last year, we were very active in the free agency market, so that week before was a huge week as far as interviewing and talking and stuff like that," Cheveldayoff said. "This year, the conversations are more about trying to get our own guys in place."
That includes looking beyond next season because of the pending contract situations, especially with Wheeler and Laine. Cheveldayoff said he and Wheeler, the Jets captain, have had an open dialogue for a while on the current and future state of the organization.
"There's been some good conversations that Blake and I have had and we'll leave it at that," he said.
Cheveldayoff wouldn't go into details about Laine either, but it's possible the Jets work out a contract with him at some point before next season.
"There's a lot of work to get done, but from the day we were able to select him (No. 2 in the 2016 NHL Draft) we knew he was a special player," Cheveldayoff said. "We're fortunate to have him and his talents. He's a big part of the success we had this year and why we feel so good about the Jets moving forward."