Both cases are politically sensitive and the decision to proceed with them has been connected with statements by Mr Chirac and his Justice Minister, Jacques Toubon, who took office a week ago. Mr Chirac pledged to uphold the independence of the judiciary. Mr Toubon promised never to use his position to halt judicial investigations and said journalists should have free rein to publish details of corruption in high places. The two cases just opened could sorely test those intentions.
The Paris case centres on Georges Perol, who retired two years ago as head of the capital's public housing.
He is an influential member of Mr Chirac's RPR party, mayor of a town, Meymac, in Mr Chirac's country fiefdom in the Correze, and was appointed to his Paris post in 1982, when Mr Chirac was mayor. They are regarded as friends.
The investigation concerns false accounting that allegedly allowed tens of millions of francs in city receipts to be siphoned off to RPR funds in the central regions of Limousin and Auvergne.
The case has similarities with the recent false-accounting case in which Henri Emmanuelli, current head and former treasurer of the Socialist Party, was found guilty of complicity, and illustrates how political appointees were able - some would say expected - to use their administrative positions to benefit their parties.
The sensitivity of the Perol case arises not just from the Chirac connection but because Mr Chirac was occupying a Paris flat bought by a housing association with help from the council, and because Mr Perol's office was nominally supervised by Jean Tiberi, who has just become Mayor of Paris in succession to Mr Chirac.
Magistrates this week also made known their intention to continue investigations into another potentially difficult case, concerning financing of the centre- right Republican Party. The case, which seemed stalled during the presidential campaign, is believed to implicate the new Economy Minister, Alain Mad- elin, who was secretary of the party in the late Eighties.
The other new investigation case concerns Guy Dejouany, chief executive of Compagnie Generale des Eaux, who has been questioned in connection with the awarding of a contract to supply water to St Denis, in the French Indian Ocean island of Reunion. Around 30 other people are also under investigation in connection with the case. Mr Dejouany becomes the latest head of a large French company to come under investigation for corruption. Two executives of the Alcatel telecommunications company have been suspended from their posts in connection with a separate case involving over-billing.