What could cause oil to be in my radiator?

I went to check my coolant level and when I removed the radiator cap I found the coolant to be yellow and oily feeling. The coolant was a nice green color a few days ago. I have been checking it often because I noticed a small dripping coolant leak coming from the top of the water pump area. It looks like maybe seeping out from the gasket, but can't tell. The coolant is also a lot thicker in consistency, something like a melted milkshake. The entire radiator is not full of oil, just a couple inches on top. I checked the motor oil and the level is about the same place on the dipstick, so I'm not missing a bunch of oil from the crankcase. There is no indication of water in the crankcase, as the oil on the dipstick is its normal brown color with no discoloration or water droplets. Is it possible that a failing water pump could somehow let oil into the coolant system?