The torch is journeying through Italy's 110 provinces ahead of the start of the Milano-Cortina games, scheduled for February 6-22.
Allegri walked with other volunteers through the city of Borgomanero, about 70 kilometres (45 miles) northwest of Milan.
Some 10,001 torchbearers have been mobilised to carry the flame, wearing white uniforms with a red-and-yellow pattern recalling the Olympic flame.
But former cross-country skiing champion Silvio Fauner is complaining that he and other Olympic medal winners have been sidelined.
"There's no respect for us champions. I consider it an incredible insult," Fauner said in an interview on Tuesday with sports daily La Gazzetta dello Sport.
"I represent 10 athletes who have won 35 Olympic medals, starting with the two gold relay teams of 1994 and 2006... We were not involved in the slightest in any Winter Olympics initiative in our country. Neither torchbearers, nor ambassadors, nor any role. Nothing," he said.
Olympics organisers said in a statement Fauner had been excluded from torchbearing duties because political office holders are disqualified.
Fauner is deputy mayor of Sappada, a ski resort in the Dolomites.
In a follow-up on Facebook, the retired athlete complained of double standards, noting that a local politician was among the torchbearers in Sicily.
He said he was speaking up for "at least 15 (other) athletes who have won Olympic medals in winter sports, champions who have written the history of Italian sport and who today feel sidelined."
Italian Infrastructure Minister Matteo Salvini, who is heavily involved in Olympics preparations, and Sports Minister Andrea Abodi announced on Wednesday an "urgent meeting" with Games organisers to deal with the controversy.
In a joint statement, they said they wanted to shed light "on very baffling decisions".