The overflow tube attaches to the flapper valve base - it's all one unit bolted into the drain hole in the bottom of the toilet tank. But it's very rare for the base itself to develop a leak, while the flapper needs replacing every couple of years as it starts to leak a little. Just buy a replacement flapper - they're cheap, and it's not like you won't need one in the future anyway.
Everyone should do a leak check on their toilets at least once a month to prevent an unexpected jump in your water bill. All you need to do is adjust your angle of view of the toilet bowl until you can see a reflected light shining off the surface of the water. Move around to see all parts of the water edge. If you see the tiniest ripple in the reflection, your toilet is leaking. Another way is to put a few drops of food dye in the tank and see if the colour shows up in the bowl, but there's really no need for that - the reflection check is quicker.