The 31-year-old forward is in the final season of a four-year contract he signed March 2, 2015, and can become an unrestricted free agent July 1.
"I don't really care about that," Zuccarello said about his contract status. "I'm here to make the playoffs and hopefully get as far as we can in the playoffs. Everything else, I don't care about right now."
Zuccarello has 37 points (11 goals, 26 assists) in 46 games this season and has led the Rangers in scoring each of the past three seasons.
"He makes great decisions with the puck, tireless worker and someone who can play in every situation," Stars coach Jim Montgomery said about Zuccarello. "He's just going to add a diversity, a creativity to our forward group that is much needed and will make us a better team."
Signed as a free agent May 26, 2010, Zuccarello has 352 points (113 goals, 239 assists) in 509 regular-season games and 31 points (11 goals, 20 assists) in 60 Stanley Cup Playoff games, all with New York.
"I mean, you've been there for nine years, so obviously it's a little bit emotional," Zuccarello said of being traded by New York. "At the end of the day, I probably knew for a long time I was going to get traded. So it was almost like, you're just happy once it was over; it was kind of a relief. Now I can just focus on the next chapter here, hopefully I can contribute with Dallas and do my best here."