Avery, known as an "agitator" for his aggressive play and controversial comments, would be eligible for demotion to the Rangers' American Hockey League affiliate in Hartford if he isn't claimed within 24 hours.
Tortorella said the Rangers have "better players" available than the 31-year-old forward.
"I don't want to jam up Sean here -- I think we have better players than Sean Avery, plain and simple," Tortorella told the New York Daily News as the team prepared to open its season Friday against the Los Angeles Kings in Stockholm.
"I can dodge it 10 different ways, without trying to run Sean over. I thought he had a good camp. But I think with the makeup of our team, and some of the people we've added, and some of the youth we've added, as far as depth, put Sean in this spot."
New York claimed Avery off waivers from Dallas in March 2009 after the Stars had signed him to a four-year, $15.5 million deal.
He notched three goals, 21 assists and 174 penalty minutes in 76 games last season.