Despite Allardyce being contracted until the end of the 2018-19 season, one fan would have Mancini "in a heartbeat", while another described him as a "tactical genius".
Mancini has beaten off competition from the likes of Carlo Ancelotti and Antonio Conte to land the gig, and will now be tasked with steering Italy back to the top of the global game following the disappointments of 2017 and the failed World Cup qualification campaign.
Mancini, former Man City and Inter Milan coach, signed for three years with Zenit last summer and led the club to a fifth-place finish in the Russian league. Mancini, who has reportedly been engaging in talks with Italy to take the helm of the first team, is to leave Zenit after their final league game in the 2017/2018 season against Khabarovsk.
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"It has nothing to do with the club or the national team", he added.
Mancini meanwhile said he would be proud to coach Italy, telling Italian radio: "If one day I got the chance to sit on the national team bench it would be something lovely".
Italy are looking for a replacement for Giampiero Ventura, who was sacked in November after the Azzurri failed to reach the World Cup for the first time in 60 years.
The Italian team will meet on May 22 to prepare three worldwide friendlies against Saudi Arabia on May 28, France on June 1 and Holland on June 4.