Vesey had 50 goals and 40 assists in three seasons with New York. He posted a career-high 35 points on 17 goals and 18 assists last season.
Vesey, 26, is under contract for next season with a cap hit of $2.275 million.
He was drafted by Nashville in 2012 out of Harvard, where he won the Hobey Baker Award. With indications that he wouldn't sign with the Predators, his rights were traded to Buffalo in June 2016. He jilted the Sabres, however, and opted to become a free agent and then signed with the Rangers.