With Italy facing the prospect of a playoff to reach the finals, Tavecchio told local media it would "go against history" if they did not make it to Russia and he had spoken to FIFA chief Gianni Infantino about reviewing the qualifying system.
Italy were beaten 3-0 by Spain in Madrid in Group G on Saturday, leaving them three points behind their rivals in second spot.
The nine group winners go directly to next year's tournament while the best eight second-placed teams play off for four more places.
Italy have been at every World Cup apart from 1930, when they did not enter, and 1958, the only time they failed to qualify.
"The idea of not qualifying for the 2018 World Cup would be an apocalypse," Tavecchio said on the sidelines of an Italian Football Federation (FIGC) meeting.
"I've spoken to Infantino about reviewing the qualifying system. It's not right that they don't take into account the four world titles; it goes against history."
"FIFA will work on this shortly," added Tavecchio, who had previously complained about Italy and Spain being drawn together in the same qualifying group.
FIFA could not immediately be reached for comment and there were no further details Tavecchio's proposals.
Italy coach Giampiero Ventura, whose side host Israel on Tuesday, said he had not contemplated failing to qualify.
"When the draw was made, we knew that being with Spain meant there was a possibility we would not qualify directly," Ventura told reporters.
"I'm still sure we can qualify for the World Cup and we are not thinking about the apocalypse, as Tavecchio calls it."
(Writing by Brian Homewood in berne; Editing by Ken Ferris)
