Out of the 35,000 cities, towns and villages that France counts, about 30,000 have already elected their mayor in the first round. But many French large cities are among the 5,000 remaining, including Paris, Lyon and Marseille.
Philippe said the elections could only be held at the end of June if conditions surrounding the coronavirus pandemic would allow it and that no final decision had been made, according to party leaders.
A postponement beyond July 1 would require cancelling the results of elections held in March for the cities that did not elect a mayor at the first round and organising a new ballot.
The first round was just two days before the start of a nationwide lockdown, to the bewilderment of many as Philippe then said the vote would go ahead even as he declared all cafes, bars, cinemas and non-essential shops would close until further notice.
France is the world's fourth worst-hit country in terms of deaths from the novel coronavirus. Its lockdown started to ease last week.
"The majority of the participants are in favour of a ballot on 28 June," the head of centrist UDI party Jean-Christophe Lagarde told Reuters, asking for a postponement of the vote to 2021.
The organisation of a second round at the end of June "is the preferred option, it seems," said Socialist Party leader Olivier Faure.
* French health authorities reported 110 new coronavirus deaths on May 20, an increase of 0.4%, bringing the total to 28,132, the fourth-highest in the world behind the US, Britain and Italy. The number of confirmed cases increased by 418 to 143,845, an increase of 0.3%, in line with the average rise per day seen since the unwinding of a national lockdown on May 11.
