Nowa Deba, in south-east Poland, has declared itself as an LGBT+ free zone along with around 100 other areas in the country.
The towns twinned in 2006 and the relationship has been strong until Nowa Deba took this position.
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Cork County Councillor Noel McCarthy says the council has written to the town calling for them to change their policy.
"We have written to the Nowa Deba municipal district to say that we are suspending at the moment, waiting for a letter, to drop what they have done.
"If they don't with back with a positive response, or they don't write back at all, we will terminate all ties with Nowa Deba because of their stance on LGBT rights."
