no it shouldn't but you will be losing water, and more than likely will be getting into the engine oil.
if you are getting water in your oil its eather a busted hear or head gasket if no water in oil it could be water pump
Usually you get oil in your radiator when you blow a head gasket and oil leaks into your radiator. Blown intake gasket is also possible, as well as a hole in the engine's water jacket (very uncommon). Also, if the vehicle's transmission cooler coil is internal to the radiator, a leak from it may be possible (most tranny coolers are external for this reason).
Oil in the radiator is usually an indication of a leaky head gasket.
Check your heater hoses, that's where the water ciruclates into the heater from the water pump.
Blown Head Gasket
It means that your engine oil is mixing in with your radiator fluid. You probably have water in your oil too. I would take it in and have it fixed.
the thick liquid would plug your radiator and ruin your water pump. and destroy your engine.
if this vehicle is automatic,the most likely cause would be the transmission fluid cooler. it is a tank inside the radiator that cools the transmission fluid. if by any reason the tank leaks it would contaminate your water/coolant/antifreeze. answer a cracked head will allow oil into water jacket in head, going from there thru to radiator if it was a bad head gasket, u would have water in oil instead
You did not say what you are working on, YEAR AND ENGINE SIZE. But in most cases or designs the water pump can not leak oil into the coolant. The water pump has nothing to do with the engine oil. Does the engine have a engine oil cooler on it ? That would be 2 oil lines just above the oil filter that runs down the side of the engine oil pan a go to the radiator. There are tubes inside of the radiator that oil flows through and they are none to leak oil into the engine coolant that's inside of the RADIATOR. If they are leaking then REPLACE RADIATOR. BUT like I said I don't know what you are working on. Would like to help you with your problem, But need more INFO. NEUTZ.
You have a blown headgasket or cracked cylinder head/engine block. It can also be a leak at the oil cooler. Its at the oil filter mounting, check that 1st.
(Oil floats) mostly mixes with the water forming emulsion which clogs the radiator.
If it is a water-filled or oil-filled radiator, it will leak fluid.
sounds like you have a cracked head or cracked block.however, i would check to make sure it is not transmission oil and thereby ruling out a radiator problem
oil in the radiator water is normally caused by a faulty inlet manifold gasket.
Pressurized oil lines go from the block through to the head. If the head gasket is leaking at that point and allowing a small amount of pressurized oil to seep between the line and the coolant jacket you could get oil in the radiator. Another possibility is that perhaps someone used water pump lubricant which is nothing more than an emulsified oil (oil blended with water). Over time the oil can separate from the coolant and that might be all you're seeing, especially if it isn't much oil. If it's RED oil it could be comming from the transmission cooler line. That runs through one of the radiator tanks and if the transmission line is leaking you could certainly get oil in the radiator... but it' would be red.