Vesey notched 17 goals and a career-high 35 points in his third year with the Rangers this past season. He has one year left on the two-year, $4.55-million deal he inked with New York last July.
The Sabres acquired the 26-year-old's negotiating rights in August 2016 from the Nashville Predators, with whom he refused to sign after being drafted four years prior by the Central Division club.
Vesey then opted not to ink a deal with Buffalo and instead joined the Rangers later that summer.
He won the Hobey Baker Award as the NCAA's top men's hockey player in 2016.