The proposal is to start the three-week race in Belfast, with one of the first stages going to Dublin. If successful, it could mean a €7m windfall.
It would also give the public here to get up close to names such as Tour de France winner Bradley Wiggins.
In 1998, the Tour de France came to Ireland, but was tainted by a doping scandal. Days before the start, an assistant with the Festina team was caught with steroids and EPO.
The 1998 tour began in Dublin with an individual time trial and was followed by stages in Dublin and from Enniscorthy to Cork.
The French newspaper L'Equipe has reported that Belfast was set to be the start city for the 2014 Giro. The race was then expected to head south of the border, with a stage to Dublin.
Belfast city council is understood to be putting in £400,000 (€466,000).
It is expected that the Giro's start, known as the Grande Partenza, would take place over the bank holiday weekend of May 2 to 4.
The official start usually takes in six days, with a three-day festival followed by the first three stages of the race proper.