Ferenc Gyurcsany, the prime minister, said on Tuesday that Hungarian secret services had received the information about the attacks from another European Union member state.
Gyurcsany did not name the country, but Nepszabadsag, a Hungarian daily, said the information came from secret services in neighbouring Slovakia.
The prime minister said those planning the attacks were three members of a well-known organised crime group.
"The Hungarian secret services had been officially informed that foreign criminal groups, with the aim of destabilising Hungary's political situation, intend to carry out explosions in Hungary," Gyurcsany told a news conference.
"The planned locations of the explosions are the campaign rallies."
Gyurcsany said the source of the information was checked and proved to be "reliable and controlled", and police will step up security at the rallies. Border controls will also be tightened.
Support for the ruling socialists and main conservative opposition Fidesz party has shown the two parties in a close race in the first round of the election on April 9.
The socialists said they would go ahead with their rally on Saturday and Fidesz holds a rally on Sunday.