"They just want to know," Florio said. "They want to know because in an NFL locker room, it's a different world. It shouldn't be that way."
Te'o only recently overcame the embarrassment of the public exposure brought when it was revealed in January that he was fooled by a male acquaintance into believing he was dating a woman he genuinely loved.
Te'o is projected to be a first-round pick and is the second-ranked inside linebacker in the draft by NFLDraftScout.com.
Teams are also digging into the reasons why he was erased so easily by Alabama's prospect-heavy offensive line in the national championship game.
"On the field, you still have to account for what happened in the BCS national championship game against Alabama," Florio said.
Florio said teams are likely not asking Te'o directly about his sexual orientation because the inquiries could get leaked to the public, much like when Miami Dolphins general manager Jeff Ireland asked then-Oklahoma State wide receiver Dez Bryant (now with the Dallas Cowboys) if his mother was a prostitute in 2010.
"I don't think you can ask that question -- because of the Dez Bryant question of a few years ago," Florio said. "If you go Katie Couric on Manti Te'o, it's going to come out that some team, some GM, some coach asked Manti Te'o if he's gay. Then you'll have to deal with all the distraction and all the stuff that flows after that."
Patrick asked Florio if teams are finding a more finesse way of asking Te'o if he is gay or if they are just doing investigative work behind the scenes.
"It's been described to me as the proverbial elephant in the room and I don't think anyone knows how to solve this dilemma yet," Florio said. "It's just that they want to know what they're getting. They want to know what issues they may be dealing with down the road. We just assumed that at some point there would be an openly gay player in an NFL locker room and the team would have to work with the realities and make sure that everything's fine.
"Now you have a situation where you have a guy who was in a relationship with a fake girl who ended up being a real man, and the man has said to Dr. Phil that he was romantically in love with Manti Te'o, it just raises a lot of questions that the NFL at some point is going to have to deal with."
Florio said former Indianapolis Colts coach Tony Dungy (now an NBC Sports analyst) told him that "if the NFL were ready for an openly gay player, there would be an openly gay player."
It is not clear if Te'o's draft status will be affected by the hoax.
"I'm not saying anyone would take Manti Te'o off the board if they suspected he's gay or know he's gay," Florio said. "That's just the thing that's out there that they want to know the answer to."