Connor, a five-year veteran, is competing with Mark Herzlich for the job. Herzlich has received the bulk of the reps with the first team, though Connor remains optimistic the starting role is within reach.
"There's no jobs given out; the players who've been around for a while and are older -- they're looking to replace them, and the younger guys are hungry," Connor said. "It's always a competition. The coaches create an environment where you know that if I work hard and produce, they'll play me.
"The linebacker position in particular -- it's an open job, anyone can go for it. It's just who will excel and make plays."
Connor spent his first four seasons with the Carolina Panthers, battling injuries but producing consistent numbers when healthy.
Last year, Connor was a member of the Dallas Cowboys, where he racked up 29 tackles in eight starts in a 3-4 defense. It was the first time in his career he played in that system. During the off-season he signed with the Giants as a free agent.
With the Giants, he returns to Perry Fewell's 4-3 scheme, one he admitted he feels more comfortable in.
"I'm more of a 4-3 guy," Connor said. "I wanted to get back into that 4-3. I'm just more comfortable playing it."
While Connor is optimistic about his chances of making up ground in the middle linebacker race, he did state he is open to playing a role on special teams. He's viewing training camp as a significant step toward earning that starting job, though.
Just as significant to what Connor can show the coaches on the field is the off-the-field incident he had in Philadelphia a few weeks ago.
Connor was arrested at Philadelphia International Airport and cited on a weapons possession charge for allegedly carrying a switchblade inside of his luggage.
When asked about the incident, Connor called it a "misunderstanding", explaining he did not know he had the pocket knife in his bags.
"It's something I have to deal with now ... Hopefully the coaches got to know that's not the kind of guy I am," Connor said. "I'm responsible for it and I have to accept any punishments handed out.
"I've got to prove to everyone that I'm not that type of guy, the type of guy to get in trouble. That was the first time I had been arrested, so it was definitely pretty embarrassing for me."