Strome, a center, is at the end of a two-year, $9 million contract. He had 54 points (21 goals, 33 assists) in 74 games this season, and nine points (two goals, seven assists) in 19 Stanley Cup Playoff games.
The Rangers were eliminated Saturday with a 2-1 loss at the Tampa Bay Lightning in Game 6 of the Eastern Conference Final.
"Who knows what happens, but I've put everything into this team and I think the guys in here have, and I would love nothing more to keep it going," Strome said Monday. "I just feel like there's a bit of unfinished business."
Strome, who turns 29 on July 11, said his agent and the Rangers were having discussions about a contract earlier in the season but mutually decided to pause those talks when the playoffs were getting closer.
Acquired in a trade with the Edmonton Oilers for forward Ryan Spooner on Nov. 16, 2018, Strome has 195 points (71 goals, 124 assists) in 263 games with New York.
"My first choice is to be a New York Ranger," Strome said. "I think I've given them an indication that's my feeling. If that's not the case than we'll cross that bridge when we come to it."
Strome said his offseason could begin with surgery to repair a core muscle injury that bothered him in the second half of the season and was reaggravated in Game 3 against the Lightning, causing him to miss Game 4 and be compromised in Games 5 and 6.
He did not return for the third period of Game 6 after playing the first two. Strome said he will find out Tuesday if he needs surgery.
"I was dealing with a pelvis injury, my whole pelvis, which makes both groins and your abs, I felt like I was playing with a knife in my abs for the last little while," Strome said. "We'll find out tomorrow [about surgery], but hopefully not. We'll see."
Strome said the surgery would not impact his availability for the start of next season.
"He's such a character guy in our locker room, a big leader," center Mika Zibanejad said. "He knows when to be serious and he knows when to get the guys to kind of loosen up a bit. He's such a big part of our team. He has been since Day One. Such a smart hockey player; incredible to bounce ideas off of and just talk really about anything. He's been a huge part of our team."
Strome played the majority of the past three seasons playing on the same line as forward Artemi Panarin. Both said their chemistry on the ice is matched by their friendship off it.
Panarin said "of course" when asked if he wants Strome back with the Rangers.
"We talk in the rink, away from the rink," Panarin said. "He's a big part of my life. I can speak with him about deep things, about my thoughts. We share with each other everything."
Strome is one of New York's seven pending UFAs, a list that also includes forwards Andrew Copp, Tyler Motte, Frank Vatrano, Kevin Rooney and Greg McKegg, and defenseman Justin Braun.
Copp, Motte, Vatrano and Braun were acquired prior to the 2022 NHL Trade Deadline on March 21. It's unlikely all of them will be back, but Copp, Motte and Vatrano also expressed an interest in returning to the Rangers. Rooney did as well.
Copp said he would have discussions in the next couple of weeks with coach Gerard Gallant and general manager Chris Drury about what his role could be moving forward. Copp had 14 points (six goals, eight assists) in 20 playoff games.
Rangers defenseman Jacob Trouba is one of Copp's closest friends and said he will be speaking with him as well about coming back.
"[We'll see] where they see me long term and go from there," Copp said. "Long term is not only just next year, but hopefully four or five years. How do they see me in the room in terms of the leadership group, all those things. A lot to take in, a lot to hear and then just make the best available decision.
"I'm not married to one thing or the other, but I definitely did enjoy my time here and it definitely feels like there is unfinished business. I really like the group and the direction the team is going."
