Smith is in the final season of a two-year contract and can become an unrestricted free agent after the season. TSN reported the Rangers are interested in re-signing him.
Red Wings general manager Ken Holland said he was in discussions with Smith about a contract extension for the past week.
"I was also very up front that we've traded away a lot of draft picks through the years, we're having a tough year, we're dead last in the [Eastern Conference] and if there was an opportunity at the [NHL Trade Deadline] to make a deal to get the assets we would do it," Holland said. "They were aware of that. That's why the negotiations were dragging out until [Tuesday]. We were making good progress. We were close on the parameters. I talked to the Rangers [on Monday], a couple other teams got in the mix, and obviously pulled the trigger today on the deal that we did."
The Rangers reportedly were pursuing defenseman Kevin Shattenkirk before he was traded to the Capitals on Monday. Losing out on Shattenkirk likely pushed New York to move quickly on Smith, who is expected to make his Rangers debut at the Boston Bruins on Thursday.
"A good player like that, he's going to go somewhere kind of expected," Gorton said of Shattenkirk. "There's going to be one or two teams if it wasn't going to be us. It sort of went the way we thought it would. … We had conversations, and at some point we decided it wasn't in our best interest to go forward."
Smith fills an immediate need with the ankle injury to defenseman Dan Girardi, who is expected to miss at least two weeks. Defenseman Kevin Klein missed his fourth consecutive game Tuesday with a back injury. Rangers coach Alain Vigneault said he hoped Klein would practice Wednesday.
Defenseman Steven Kampfer was recalled from the American Hockey League on Tuesday and played 15:44 against Washington.
Smith, who is a left-handed shot known for playing on his off side, could get the opportunity to play on the right side of the Rangers' top defense pair with Ryan McDonagh. They played together for three seasons at the University of Wisconsin from 2007-10. Rangers center Derek Stepan also played at Wisconsin from 2008-10.
Holland said getting draft picks was key for him. The Red Wings also got a third-round pick this year from the Chicago Blackhawks in a trade for forward Tomas Jurco on Friday and will get the Toronto Maple Leafs' third-round pick as compensation for coach Mike Babcock, Holland said.