PARIS (Reuters) - Marine Le Pen's far-right National Rally came third in France's snap legislative elections this summer. Two months later, though, they are now the power behind the throne, propping up Prime Minister Michel Barnier's government.

President Emmanuel Macron named former Brexit negotiator Barnier as prime minister on Thursday, capping a weeks-long search following his ill-fated decision to call a snap legislative election that delivered an unruly hung parliament.

Yet the fates of Barnier - and Macron - now rest with Marine Le Pen's far-right National Rally (RN), opposition lawmakers and analysts said.

"Today, we have a prime minister who is completely dependent on the RN," said Lucie Castets - the prime ministerial pick of the leftist alliance that came top in the July vote, but who was overlooked by Macron - on Friday.

"In so doing, the president has put himself in cohabitation with the RN," she said, referring to the rare political phenomenon in which French presidents occasionally govern with a prime minister from a rival party.

Macron took weeks to name a prime minister as he was looking for someone who would not immediately be toppled, and who would preserve his legislative achievements - requirements that gave the RN outsized power over the selection process.

The RN gave tentative support to Barnier's nomination on Thursday, but made clear it could withdraw support at any point if its concerns on immigration, security and pocketbook issues are not met.

"We reserve all political means of action if this is not the case in the coming weeks," RN party president Jordan Bardella said on Thursday.

French Green party leader Marine Tondelier said Macron had given Le Pen the power of Julius Caesar overseeing a gladiatorial battle, giving a thumbs up or down to his candidate for prime minister.

"It was her approval that was sought," she said on TF1.

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Socialist Party president Olivier Faure said Barnier would be fully aware his fate rested in the hands of Le Pen, adding that nobody from his party would join the government.

"It is now the extreme right that makes the kings or queens," he said on France Inter.

RN lawmaker Laurent Jacobelli sought to play down the far right's power over the new government.

"We are in the opposition, clearly ... We are not giving anyone the green light," he said on France 2. "We will not be the political force that leads France into the wall by systematically opposing everything and creating chaos."

Much attention will now focus on who Barnier seeks to name to his cabinet, with the risk of protests, called by the left for Saturday, also on the horizon.

Eurasia Group, a political analysis firm, said on Thursday some ministers from the outgoing government may continue but Finance Minister Bruno Le Maire was unlikely to be among them.

"Barnier's choice for the finance ministry will therefore be crucial to reassure Brussels and financial markets that France can emerge safely from a dangerously intertwined political and fiscal crisis over the next three months," the firm said.

Eurasia also highlighted the power of the RN over Barnier.

"The key figure to Barnier's hopes of success - or in the short term survival - will be the far right's Marine Le Pen," it said.

(Reporting by Gabriel Stargardter; Editing by Benoit Van Overstraeten)

By Blandine Henault and Benoit Van Overstraeten