One tunnel? Sure.
Put the exit into a place the infiltrators will hold, or a semi-public place they can infiltrate. The basement of a neutral merchant's home-office-warehouse, say, or the basement of the university. Brick it over until needed and put a jumble of old stuff there that is too good to throw away, to worn to use. Rickety chairs, chipped mugs, whatever.
In the best case an agent in place will secure and clear the exit. The merchant mentioned above, or the chief clerk of that merchant. A janitor for the university. Otherwise they clear it from the tunnel, more noisy but it should be possible.
The tunnel would only be found if one of the exits is found, or if new construction accidentally hits the tunnel. So go deep enough.
Many tunnels? More difficult.
With many tunnels, the odds of accidental discovery go up. Bad enough if the tunnels are not connected, since it would cause awareness of hidden tunnels and possibly a systematic search. If the tunnels are connected, the whole system is compromised.
Why dig secret tunnels? The city has legitimate sewers, and all you need are secret bypasses for the places where those sewers pass under the walls. That's assuming there are big iron grates or something like that which are inspected periodically. The bypass has several doglegs and does not appear to go under the walls. It may or may not be bricked up.
Explain why they don't use the gates.
Not even a siege of a city that large will be spy-proof. If it is less than a siege, there will be thousands of legimitate travelers through the gates every day, and not even modern technology could make 100% forgery-proof papers.
What are criminal smugglers using? If they know the tunnels (or use similar ones) that could be problematic, but it would also reduce the outcry if one of yours is found. "Yet another whiskey-smugglers' tunnel. Tell customs and let them handle it."