Well, there's a lot of ways this can happen. Are you able to get up on the roof? Is there water in the attic too? It could be flashing. Depending how it's constructed, it can be the failure of any joint. If you can't get on the roof, I'd get a chimney company out there to check it for you-------or even a roofer could inspect this area.

Are you certain that it is coming down the flue? On an older home that has an exterior chimney for a furnace flue oftentimes there is a cleanout inside the basement under the thimble ( where the exhaust enters the chimney) for the furnace, quite often this cleanout is below grade and if the chimney or foundation is in poor shape in this area, ground or surface water will escape into the basement through the cleanout, making it look as though the rain is coming down the flue pipe.( the same thing can happen if there is no cleanout but the thimble is below grade) This can also be made worse if there is a sidewalk or driveway against the chimney that tilts towards the chimney & not away from it.
Hope this helps
Martin

High Heat Sylicone/Mortor Patch is your friend. Or call a Sweep or Chimney Service to come inspect it as the last person suggested. No offense, but it sounds like it could use the inspection anyway.

Good Luck.