Kucherov scored 30 goals and led the Lightning with 66 points last season. He also led them with 11 goals and 19 points in the Stanley Cup Playoffs, and has one goal in two games playing for Team Russia at the World Cup of Hockey 2016.
"Obviously we want him here," Yzerman said. "He's a tremendous player, he's a popular player on the team. Being young he is still one of the leaders on our team so we want to get him in here as soon as possible. It takes time. Does it become a distraction? I don't anticipate that. Again our goal is to have him signed and in here as soon as possible. We'll deal with everything one step [at a time] but it's our hope to have him signed and in camp as soon as he's done [with Team Russia] but that may or may not happen right away."
Kucherov, along with defenseman Nikita Nesterov, are Tampa Bay's remaining unsigned restricted free agents.
"I think [Kucherov's] importance [to this team] speaks for itself," forward Ryan Callahan said. "The playoffs he had two years ago, and then last year how good of a regular season he had and another playoffs. He one of the main reasons we've had two really good years here. There's no hiding the fact he's an important piece."
Kucherov has 42 points (20 goals) in 45 playoff games the past two seasons. The Lightning made the Stanley Cup Final in 2015 and the Eastern Conference Final last season.
"He's a very talented player," forward Brian Boyle said. "He's a hard worker and he's an honest player. He's always had plus seasons here and he continues to get better. He's going to be a major part of this team."
Callahan, who had hip surgery in June, said he is on pace to return in November. The expected recovery at the time of surgery was five months. The Lightning open the season against the Detroit Red Wings at Amalie Arena on Oct. 13.
"For the most part I think we all generally want to be here," said forward Tyler Johnson, who is fully recovered from a wrist injury he sustained during the 2014-15 season. "I don't see anyone on our team saying, 'Hey, I want to get out of Tampa,' so we don't worry about that. This is an unbelievable place to play, unbelievable ownership and obviously playing in Florida is a pretty sweet spot."
