The deal is worth $20 million with an average annual value of $5 million, according to multiple reports.
Last year's feel-good story, Robin Lehner -- who overcame mental health and substance abuse issues to become a Vezina Trophy finalist -- signed a one-year deal with Chicago.
"We did everything we could earlier, like we did with all the other players," Islanders general manager Lou Lamoriello said. "Unfortunately, it did not work out."
The Islanders had some good news for their fans when they announced that captain Anders Lee has re-signed.
Varlamov went 20-19-9 for the Avalanche last season, splitting time with 2018 free-agent signee Philipp Grubauer. But Varlamov was on the bench in the postseason as Grubauer helped Colorado reach the conference semifinals.
Before that, Varlamov said in March that he hoped to return to the Avalanche next season.
"I want to stay here because I love this team," he told The Athletic at the time. "Hopefully I am going to stay, and hopefully I'm going to re-sign my next contract with the Avalanche."
Varlamov, 31, is an 11-year NHL veteran with the Washington Capitals and Avalanche. He has a career save percentage of .916 and goals-against average of 2.68.
While being healthy last season, Varlamov missed the end of 2017-18 with a lower-body injury and underwent hip surgery in April 2017.
Information from The Associated Press was used in this report.