Bathroom Sink Repair

I have late 40's house with the original sinks in both bathrooms. These are really heavy-duty metal sinks colored white (covered in whatever they cover sinks with). Overall they are in great shape and even the original fixtures are in good shape. These thing sare very well built and I would prefer not having to replace them as they are attached to the wall and the tiling is done around them.
The only problem I have is that where the overflow drain holes are, rust is coming out into the visible part of the sink. My question is, is there anything I can do to address this? I was thinking of using some sort of grinder to go down to bare metal right around where the rust is and then using some sort of enamel/sink repair product to fix it.
Would this work or do I need to tear the sinks out and re-tile the bathroom?
Thanks in advance for any advice.
Rob
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Rob wrote:

Sinks like yours are cast iron covered with porcelain. Basically, the iron is heated hot enough to melt glass (porcelain) and then dusted with porcelain powder.
I don't know if porcelain is applied to the interior cavity of the sinks or not but it certainly is around the overflow inlet holes. Is it still there? Or do you mean that rust exits the hole area whan the sink is full?
As far as fixing, I doubt that it can be done. Yes, the edges of the holes could be done to some degree but you couldn't reach into the interior cavity and rust there would inevitably advance again. If the rust just comes out the holes when the sink is filled to overflowing the simplest thing is just to not fill it that full.
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dadiOH
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dadiOH wrote:

Thanks for the response. The rust is advancing from the interior of the overflow drain into the visible part of the sink, it is not just staining....
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