Amandine Gilles, 33, and Angélique Leroux, 27, hope to be married in their hometown in south-eastern France on 10 September.
But on Friday Bompard met them and told them that she would not carry out the ceremony because it was against her religious principles.
She would also refuse to delegate another councillor to carry out the ceremony, as the law allows her to do.
Bompard is a member of far-right split from Marine Le Pen's Front National, the Ligue du Sud (Southern League), led by her husband, Jacques.
Jacques Bompard, who is a member of the city of Orange, has also refused to carry out a gay marriage but delegated his power to a fellow councillor.
Accusing the government of threatening "disproportionate and illegitimate" sanctions against mayors who refuse to carry out the gay marriage law, Bollène council's ead of communications, Jean Vallier, insisted that the mayor's opposition was a "case of conscience".
The mayor of nearby Saint-Saturnin-lès-Avignon, Bernard Goudon, phoned the couple to offer to carry out the ceremony but they have written to the region's prefect to call on him to demand that Marie-Claude Bompard carry out her legal obligations.
Bompard said earlier that she would ask the prefect to designate somebody else to carry out the ceremony.
Gilles and Leroux told the Vaucluse Matin newspaper that they have received dozens of expressions of support from local people, starting with the post office employee to whom they handed their letters to the prefect and the mayor on Saturday.