Brown is waiting on results from a specialist and if an operation is necessary, it would likely mean him missing the second leg of the Champions League tie at Juventus on March 6 and Scotland's World Cup qualifying double-header against Wales and Serbia later next month.
Ahead of the visit of Dundee in the SPL on Sunday, Lennon said: "The specialist is coming back with his prognosis. He may need surgery but we won't know that until Monday, the specialist is on holiday at the minute.
"We are expecting the specialist to say he may need surgery so as soon as we get him a slot, if that is the case, then we will get it done.
"He is a very important player, the captain, but the squad is strong."
Lennon also revealed that Thomas Rogne has rejected the offer of a new deal. The Norwegian defender's contract is up at the end of the season.
"We have made him an offer and it has been refused," Lennon said. "We may go back to him with another offer but there are no guarantees on that.
"Obviously we don't want it to fester on, we would like to get a result one way or the other.
"Thomas is a good player, we would like to keep him around and there is a good chance of him playing on Sunday.
"He has had to be patient because Efe Ambrose, Charlie Mulgrew and Kelvin Wilson have done exceptionally well, but whenever Thomas has come in he has never let us down."
On midfielder Paddy McCourt, whose current deal also expires in the summer, Lennon said: "We will talk to Paddy and his representatives over the next couple of months."