In 42 games with the Rangers this season, Del Zotto registered two goals and nine assists for 11 points.
The native of Stouffville, Ontario, has played in 292 career NHL regular-season games, all with the Rangers, and has accumulated 26 goals and 95 assists for 121 points.
He was originally drafted 20th overall by the Rangers in the 2008 NHL Entry Draft.
Klein, 29, tallied one goal, two assists and 21 penalty minutes in 47 games with the Predators this season.
The native of Kitchener, Ontario, has played in 403 career NHL regular-season games, all with Nashville, and has accumulated 16 goals and 66 assists for 82 points.
He was originally drafted by the Predators in the second-round (37th overall) of the 2003 NHL Entry Draft.
Klein is in the first-year of a five-year, $14.5 million contract, while Del Zotto is scheduled to become a restricted free agent July 1.