PARIS (Reuters) - An opinion poll published on Sunday showed that French far-right leader Marine Le Pen would lose to far-left leader Jean-Luc Melenchon in her home district in parliamentary elections next month.

The Ifop-Fiducial poll showed that Le Pen would win the June 10 first round with 34 percent of the votes against 29 percent for Melenchon but would be beaten 55-45 percent by Melenchon in the June 17 second round.

Not winning the parliamentary seat in her political backyard, the northern working-class town of Henin-Beaumont, would be a humiliation for Le Pen, who won 17.9 percent of the votes in the first round of presidential elections in April, third behind the winner, Francois Hollande, and his presidential predecessor Nicolas Sarkozy.

For Melenchon, challenging Le Pen in Henin-Beaumont - where her National Front won 35 percent in the first round of the presidential election - is a bid to remain politically relevant after he finished well behind Le Pen in the presidential election with 11.1 percent of the vote.

The Ifop poll, published by the Sunday paper "Le Journal du Dimanche", also showed that Socialist candidate Philippe Kermel would win 18 percent of the first-round vote in Henin-Beaumont and 56 percent if he were to make it into the second round.

Kermel is expected to withdraw from the second round if Melenchon has the higher first-round score and vice versa. The Ifop poll did not test the unlikely scenario of a disagreement on the left, with Melenchon and Kermel both running against Le Pen in the second round, which would improve Le Pen's chances.

Unlike the presidential election, where only the two highest-placed candidates go through to the second round, in the parliamentary election any candidate backed by more than 12.5 percent of the registered voters goes to the second round, which often has three and sometimes four candidates in some districts.

Le Pen played down the Ifop poll findings.

"The polling institutes should be a bit more modest, in particular Ifop, which forecast that Melenchon and I would have an equal score in the presidential election, while I ended up with 18 percent and he with 11. Let's wait for the election," she told France 3 television.

Opinion polls show that Hollande's Socialist Party will trounce Sarkozy's conservative UMP party in next month's parliamentary elections.

They show that Hollande's Socialists and other left-wing parties could together win 45-46 percent of the vote in the June 10 first round of the election, compared with about one-third for the UMP.

The left aims to build on its momentum after voters swept Hollande to victory and the presidency on May 6, bringing the Socialists back to power after a decade in opposition.