The quirky shows, which have been popular podcasts, were launched in 1995 and were created by presenter Ira Glass chronicling everyday tales and eccentricities.
Now a selection of 13 episodes is to be aired by digital station Radio 4 Extra from Sunday, May 4.
The programmes have been broadcast on 587 stations in the US, and the podcasts are downloaded up to a million times a week.
The upcoming shows focus on a man who believes he is turning into a cockroach, a day in the life of a US diner and features on babysitting and blackjack.
Caroline Raphael, the commissioning editor for 4 Extra, said it promised to take listeners into the "lives of ordinary and extraordinary people and situations".