PARIS, Jun. 23 (Reuters) - Thousands of women took to the streets in cities around France on Sunday to protest against Marine Le Pen's far-right National Rally, as polls indicated the party could win the upcoming parliamentary elections.

About 200 women's rights groups and unions organised the marches in dozens of cities, including Paris, saying women's rights come under attack when countries are governed by far-right parties. In Paris, more than 10,000 women demonstrated peacefully, organisers said.

In March, France enshrined the right to abortion in its constitution, a world first, but some RN lawmakers had opposed the legislation, raising concerns among some of the public about the party's attitudes to women's rights.

"During the debates around making abortion a constitutional right, we could well observe how the far-right deputies were very uncomfortable with the subject, they were calling for filling the cribs with French babies," Shirley Wirden, officer in charge of women's rights at the French Communist Party, said as she took part in Sunday's protest in Paris.

The National Rally (RN) party and its allies are seen coming out top in the first round of French parliamentary elections due to take place on June 30, with 35.5% of the vote, according to a poll published on Sunday.

The Ipsos survey - conducted for Le Parisien newspaper and Radio France on June 19-20 - showed the left-wing New Popular Front (NPF) alliance in second place with 29.5% of the vote. President Emmanuel Macron's centrist alliance was seen in third place, winning 19.5% of votes.

Macron called the snap parliamentary election after the National Rally came out top in European Union elections this month, with about 32% of the vote, trouncing Macron's centrist alliance (15%). The RN secured 30% of the female vote, a 10 point rise versus the 2019 EU elections.