Cospito, 55, is an Italian anarchist who has been on a hunger strike for more than 100 days to protest against being jailed under the strict "41 bis" isolation regime.
"We are raising security in all of our embassies and consulates because at the moment international anarchists are mobilised against the Italian state," Tajani told a news conference in Rome.
In December, a Greek anarchist group claimed responsibility for an arson attack outside an Italian diplomat's home, calling it an act of solidarity with Cospito.
Tajani called it the most serious incident to date but reported that numerous other attacks, acts of vandalism and demonstrations have taken place since November.
Tajani said they believe the network includes both Italians and anarchists from other countries acting in concert.
He referred to graffiti scrawled in Catalan on the building housing the Italian consulate in Barcelona.
Italian embassies, consulates or culture institutes have been targeted in Bolivia, Chile, Argentina, Brazil, Spain, Germany and Switzerland, he said.
On Monday, Cospito was moved from a prison in Sardinia to one in Milan with better healthcare facilities, as had been asked by Italy's ombudsman for prisoners.
The prisoner, who has lost more than 40kg and is reportedly so weak that he struggles to walk and keep warm, is surviving on water, sugar and honey.
Cospito is serving a 10-year sentence for shooting in the leg a nuclear energy manager for a state-controlled company in 2012 and 20 years for a series of dynamite attacks in Italy.
Prime Minister Giorgia Meloni's government has ruled out easing his detention regime, saying it cannot give in to threats or acts of terrorism.
The "41 bis", normally reserved for top Mafia bosses, is designed to prevent inmates from communicating with affiliates outside.
Cospito was placed under the regime in May, after he wrote articles from prison calling on fellow anarchists to continue their armed struggle.
Cospito has been sentenced to 30 years' imprisonment but prosecutors have appealed for it to be made a life term, with no parole possibilities.
with AP