Bach told international media he was happy to join French President Emmanuel Macron, Paris Mayor Anne Hidalgo and Olympic Games executive Christophe Dubi for a dip in the river.
"If I receive an invitation I will be very happy to join them. I hope that water won't be too cold," Bach told a news conference on Friday.
French charity Surfrider Foundation raised concerns over the quality of water in April, saying athletes competing in triathlon, marathon swimming and paratriathlon events would be running "significant" health risks due to pollution of fecal origin.