The Rangers made the move Wednesday on the second day of NHL free agency. The 28-year-old Bourque played in Russia and Switzerland last season.
Bourque has two career goals and eight points in 51 NHL games over parts of four seasons with Boston, Pittsburgh and Washington. He scored a goal and had three assists in 18 games during the 2012-13 for his hometown Bruins, his father's former team.
The Rangers were among the busy teams when the market opened Tuesday, beating out other suitors for highly coveted defenceman Dan Boyle, as they tried to make up for the loss of some key players from their Eastern Conference championship team.