The countries' prime ministers announced the bid after meeting in the Slovak town of Poprad, at the foot of the Tatra Mountains where alpine skiing disciplines would be held to make up for a lack of suitable terrain in mostly flat Poland.
"We have a clear vision. We are not interested in any megalomania," Slovak Prime Minister Robert Fico said in a televised news conference with his counterpart Donald Tusk.
The Games would build on Poland's Euro 2012, co-hosted with Ukraine, and Slovakia's experience with hosting the ice hockey world championships in 2011.The historic southern Polish city of Krakow would be the centre for the Games.
Germany's Munich, beaten by South Korea's Pyeongchang for the 2018 Winter Olympics, is a possible candidate while a planned Swiss candidacy of Davos and St Moritz was rejected by local citizens.
Fico brushed aside questions about whether the costs of hosting the games would be justified, especially as the euro zone struggles with its lengthy debt crisis.
"We have crisis now. Should we crawl under the table? When we have crisis, we should not think about the future, ignore any chances that come up?" Fico said.
Slovak investors have upgraded ski lifts and services at the Chopok and Jasna ski resort in the Tatras, drawing in tourists from surrounding countries but more infrastructure investment would be needed to prepare the region for the games' visitors.
Poprad unsuccessfully bid for the 2002 and 2006 Winter Games.