Vanek, 30, is in the final-year of a seven-year, $50 million contract and is scheduled to become an unrestricted free agent July 1.
He plans on testing the free agent market this summer, which has some teams concerned.
However, the Anaheim Ducks and Boston Bruins are both aggressively pursuing Vanek, multiple sources told TFP on Tuesday.
Two other teams are also believed to be in the mix.
The Los Angeles Kings and San Jose Sharks have also shown interest in Vanek, though it's unclear if they are still considered contenders for his services.
In addition to Vanek, the Islanders are also gauging the best possible return for defenseman Andrew MacDonald.
MacDonald, 27, is also a pending unrestricted free agent and will be traded by the deadline.
The Bruins are also interested in MacDonald, as are the Kings, Sharks, Columbus Blue Jackets and Tampa Bay Lightning, among others.