But the man, identified only as Franco, declined, explaining apologetically: "I can't, because I stink."
"I took him to dinner anyway," Archbishop Krajewski told La Stampa newspaper. "While we were eating, he explained that it was not difficult for the homeless in Rome to find something to eat, but what they lacked was a place to wash."
Asked if the sight of homeless people queuing for showers might put off tourists, the Polish prelate said: "Throughout the history of Rome, the poor have always gathered around churches."
There was a mixed response to the papal initiative from a group of homeless people camping out in St Peter's Square
"I think it's a great thing. At the moment we have to go a long way out of Rome to take a shower twice a week. We go by train but can't afford tickets so we're often thrown off," said Zbislaw, a 60-year-old Polish man living rough in Rome.
But his friend, Grzegorz Bialas, 50, also from Poland, said: "After a couple of months you'll have 500 homeless queuing to use the showers. This is a holy place - the showers should be put somewhere else."