Italy lost 4-1 a penalty shoot-out after drawing 1-1 in Zenica, and became the first World Cup winners to miss three consecutive editions of the tournament, which this year will be held in the United States, Canada and Mexico.
"I've asked him to stay and asked (Gianluigi) Buffon to stay in charge of the football side," Gabriele Gravina, president of the Italian Football Federation (FIGC), told reporters.
Buffon has a role within the FIGC which involves overseeing the senior and youth national teams.
Asked whether he would accept Gravina's invitation to stay, Gattuso repeated what he said to Italian broadcaster RAI by saying it "wasn't important".
"The defeat hurts too much," said Gattuso.
A World Cup winner as a player in 2006, Gattuso was hired in June last year to replace the sacked Luciano Spalletti, with a contract which expires after the World Cup.
Gravina also said he would not step down, but his presence as the head of the FIGC would be discussed at a board meeting next week.
