Cashman described the call as part of a "slow process" and said he's "reaching out to all agents" representing players on the market. The general manager said no offer was made to Wilson's agent, Bob Garber.
Wilson, 30, is considered the top free-agent pitcher on the market. The left-hander went 16-7 with a 2.94 ERA for the Texas Rangers this past season, though he struggled at times in the postseason.
Wilson's agent told ESPNNewYork.com's Andrew Marchand on Friday that Wilson and his gregarious personality would "thrive" in New York.
"He is one of those guys who would fit in there very well," Garber said. "He would fit in most places pretty well. New York, I think he would thrive in a city like that.
"His No. 1 priority is that he wants to play for a winning team. The Yankees have as good or better a reputation of that than anybody. It is very early in the process. We're just waiting and sitting back to see how it plays out."
A source told ESPNNewYork.com's Wallace Matthews that the Yankees may be hesitant to go after Oswalt because of his back issues.
Cashman also will monitor free-agent pitchers Mark Buehrle of the Chicago White Sox and the St. Louis Cardinals' Edwin Jackson. He may bid for Japanese star Yu Darvish, though the Yankees have made a few missteps when acquiring pitchers from overseas in the past (Kei Igawa, Hideki Irabu).
The Yankees also might try to retain Freddy Garcia, who went 12-8 with a 3.62 ERA with the club in 2011.
Cashman, who signed a three-year contract to remain with the Yankees last week, stressed after he officially returned that the Yankees' top offseason priority is pitching but that he would take a "conservative" approach.
The Yankees agreed to a five-year deal with left-handed ace CC Sabathia last week before he could opt out of his contract.
As currently constituted, the rotation behind Sabathia would include right-handers Ivan Nova, A.J. Burnett and Phil Hughes. It is unclear at this point who would emerge as the fifth starter with Garcia a free agent. Hector Noesi, who pitched well in a long-relief role last season, is a candidate.
Prospects Dellin Betances and Manny Banuelos are expected to begin the season in the minor leagues.
Ian Begley is a regular contributor to ESPNNewYork.com.