The group, known as the Fans of the Basque Beret, gave the pope the black hat on Wednesday during a meeting with the faithful at the Vatican.
As the pope mobile passed through the crowd, a member of the Basque group threw the beret to the pope who caught it, local daily Diario Vasco reported on Thursday.
"Is it for me?" the pope asked on receiving the headwear known in the Basque region of Spain as a "txapela".
"It was difficult but we achieved our goal. He (the Pope) didn't put it on properly but we don't care," said Borja Bilbao, who tossed the beret at the pontiff.
The idea to give Pope Francis a txapela came up during monthly meetings of the beret fan group.
The group formed in 1995 to promote the wearing of the item still commonly worn in the Basque Country.
Basque berets are round and have a stalk, or txortena, in the centre.
The name txapela means "the holder of (something)" and comes from a Basque tradition which sees the winners of cultural and sporting traditions wearing a beret.