Coach Tom Coughlin revealed that Tuck had surgery on his shoulder to remove scar tissue, and Canty had surgery on his knee. Defensive tackle Linval Joseph also had surgery on his ankle.
Coughlin said he guessed there is a seven-month minimum recovery for the two tight ends, meaning they could start the season on the physically unable to perform list.
"They are always a concern, because look at the date," Coughlin said at his NFL scouting combine news conference. "Hopefully we can get them back fast. Linval should be OK. It is a matter of some bone chips or something like that."
Coughlin said that Tuck feels better than he did before the surgery.
"I think it was just scar tissue," he said of the procedure. "No labrum."
After hearing what Coughlin said, Tuck tweeted: "Coach C spilled the beans. Yes I had shoulder surg. Nothing major. Will be more than ready to defend the title."
As for Canty, Coughlin didn't offer much detail. When asked if it was a major injury, Coughlin replied, "Anytime you have surgery, it is major."
But when asked if he had any reason to believe that Canty won't be ready for the season opener, Coughlin replied, "As I stand here, no."
Tuck had battled through injuries since the preseason, starting with what was described as a stinger. He missed four games early in the season and had to deal with neck, groin, toe and shoulder injuries. Tuck still finished the season with 5½ sacks in his last six games.
Canty and Joseph, in his first year as a starter, played in every game this season.
Coughlin has several other Giants on the mend. Left tackle David Diehl had surgery on a finger on his left hand but says he will be ready for the season. Coughlin also has two key defensive members recovering from last year's ACL surgeries in cornerback Terrell Thomas and middle linebacker Jonathan Goff.
Coughlin said the two unrestricted free agents have enough time to be ready for the season if they return. And he hopes to have both back.
"All you can do is you have a plan," he said. "And the plan certainly does include those two guys that you just mentioned. How soon or whatever, that remains to be seen."
Running back Ahmad Bradshaw, who played with a fracture in his foot in the final nine games of the season, is not expected to need any offseason surgery, according to Coughlin.
Ohm Youngmisuk covers the Giants for ESPNNewYork.com.