That situation has got danger written all over it. I'm not fan of unvented gas fireplaces, for safety reasons. If there's any fault in installation or operation, a carbon monoxide condition could develop, which could be toxic. In your case, maybe it's installed into an existing brick fireplace? This would further complicated a diagnosis.
Gas fireplaces develop a lot of moisture. If you have a cold, brick chimney and the fireplace is running, you might find a lot of water dripping down because of condensation. This might be in addition to your leak, or might actually BE what you assume is a leak.
There are a variety of ways to keep chimneys from leaking when you're talking about brick fireplaces, but with gas, I don't want to give advice until you've had it reviewed by a true, on-site expert.
With that said, hire an experienced, professional home inspector to inspect your fireplace - NOT a chimney sweep. Chimney sweeps, in my experience, are not trained or qualified to spot real problems, and often report minor issues as needing expert repairs only to line their own pockets. You need to get to bottom of this before any further modifications can be made - and a qualified home inspector can help you do that.