Barzal, the Islanders' top center, scored 17 points (five goals, 12 assists) in 22 postseason games after he led them with 60 points (19 goals, 41 assists) in 68 games before the regular season was paused March 12 due to concerns surrounding the coronavirus.
Defensemen Ryan Pulock and Devon Toews also can became restricted free agents.
"Right now, all I'm doing is focusing in on communication with each and every one of our restricted free agents and also unrestricted free agents and seeing exactly what we can get done," general manager Lou Lamoriello said Tuesday. "We did not have any communication during the playoffs intentionally, and right now we're talking. I feel that we'll get all our players signed in due time."
The Islanders, who lost to the Tampa Bay Lightning in the Eastern Conference Final in six games, also have five players who can become unrestricted free agents: goalie Thomas Greiss, defenseman Andy Greene and forwards Matt Martin, Derick Brassard and Tom Kuhnhackl.
Greene was acquired from the New Jersey Devils for a second-round pick at the 2021 NHL Draft on Feb. 16. He replaced Johnny Boychuk after the defenseman was injured in Game 1 of the Stanley Cup Qualifiers against the Florida Panthers on Aug. 1 and scored four points (two goals, two assists) in 21 postseason games.
"Right now, we're focusing on trying to keep our team together," Lamoriello said. " … We'd love to have Andy back. Andy did a great job for us."
Greiss, who has been with the Islanders for five seasons and shut out the Philadelphia Flyers in Game 7 of the second round on Sept. 5, is the most unlikely to return after goalie Ilya Sorokin agreed to terms on a one-year contract for next season July 14.
"I think our goaltending situation is pretty obvious," Lamoriello said. "We certainly would like to sign and have our free agents back; that has been communicated to them. We have a period of time that we have to make some intel decisions and have to make some moves. Nothing specific at this point; when it happens, you'll all hear about it.
"But the contribution that they gave was just fantastic and the quality of people that they are, so we would certainly like them back. We'll do everything we can."
With Barzal, Pulock and Toews each needing a new contract, Lamoriello did not rule out the possibility of trading salary to create extra cap room, even if it's a move he said he'd rather not make.
"Right now, we're certainly exploring each and every situation that potentially can take place," Lamoriello said. "But the last resort, if there is such a word, would be doing something certainly that you'd rather not do. But I'm reasonably comfortable that we'll be able to accomplish what we want to accomplish and have the team that we want to put on the ice once the season starts."
