Former England defender Campbell last week revealed he was "seriously considering" joining the Hoops and had spoken to Lennon about a possible move.
But with the latter not yet confirmed as permanent manager at Parkhead, Campbell is biding his time.
"I'm just waiting for Celtic to settle the situation with Neil," the 35-year-old told the Scottish Sun.
"When we last spoke, he was waiting to see if he was getting the job. I'd have to make a trip to Glasgow and have a look around, see what's happening. I will keep my options open.
"I'm not going to rush into anything."
Campbell, who is out of contract at Arsenal, has previously stated he would prefer to sign a new deal with the Gunners.
He added: "I am happy to wait and give Neil a chance and I will show respect to Arsenal and Celtic."
Meanwhile, Rangers midfielder Lee McCulloch has revealed Bullard has been phoning him to sound him out about Celtic.
Tigers star Bullard, 31, has been linked with a summer move to Parkhead, possibly in an initial year-long loan.
McCulloch said: "Jimmy's been on the phone quite a lot and he's asked me what it's like. He's been asking what life is like in Glasgow and if he would enjoy it.
"I've told him he would be idolised at Parkhead. The Celtic fans would take to him the second he arrived."