58-year-old Lille Mayor Martine Aubry won against former presidential candidate Ms Royal by a narrow margin of 42 votes.
However, Ms Royal has said she will not accept the council’s findings, because it has a pro-Aubry majority and so is likely to back her. AFP news agency quoted her as saying: "I am not going to take this."
Ms Aubry denied there was any vote-tampering and has rejected Ms Royal's proposal that the voting should be done again.
The French Socialist Party has had problems and infighting between those who want to move towards the political centre against those who want to hang on to more traditional leftist ideas.
This has caused speculation that the Socialist Party may split into two.
