The premium investors demand to hold French government bonds fell from a 12-year high on Monday as analysts said a hung parliament remained the base case, raising hopes for a limited increase in fiscal spending that would support France's debt sustainability.

Marine Le Pen's far-right National Rally (RN) party scored historic gains to win the first round of France's parliamentary elections, but the final outcome will depend on days of alliance-building before next week's run-off vote.

Political parties rushed to build a united front on Monday aimed at blocking RN's path to government.

There was a feeling of relief on the financial markets, but "we are not out of the woods yet," Alex Everett, investment manager at abrdn said.

"The National Rally (RN) exceeded expectations, and may yet pick up the second round votes for a relative or even absolute majority," he added.

The debt risk premium for other euro area countries also fell as investors saw France as less likely to endanger the stability of the bloc.

German safe-haven Bunds underperformed their peers after the vote, tightening spreads across the euro area. Bond yields move inversely with prices.

The gap between French and German 10-year sovereign bond yields - a gauge for the risk premium investors demand to hold French bonds – tightened to 75 basis points (bps), after hitting 85.2 on Friday, its highest level since July 2012. It was less than 50 bps the days before Macron called for snap elections.

France's 10-year government bond yield hit a fresh 8-month high at 3.344%, and was last up 4 bps.

"Given the division in the French parliament, we find it unlikely that the new government can find support for any larger increases in spending," Rune Thyge Johansen, euro area economist at Danske Bank, said.

Other yield gaps tightened, with Italy and Greece down 7 bps each at 150 and 114, respectively, while Portugal fell 4 bps to 69 and Spain 1 bp to 85.50. Austria and Belgium's spreads tightened by about 3 bps.

"In any case, France will likely remain politically unstable after the election, with very limited policy visibility and a substantial loss of influence in Europe, at a critical time for the continent," Citi economists said.

The exit polls aligned with opinion surveys before the election and provided little clarity on whether the eurosceptic RN can form a government to "cohabit" with the pro-EU Macron after next Sunday's run-off.

The RN's chances of winning power will depend on the political deal-making made by its rivals over the coming days.

Citi analysts said last week the French yield gap would tighten to 70-75 bps if RN leads the government, implementing just part of its fiscal plans, and would widen to 100-105 bps if RN carries out most of its budgetary goals.

Such a backdrop would affect the debt risk premium in Italy, the bloc's most vulnerable country, as credit rating agencies see an expansionary debt path. The gap could reach 140 bps or 155 bps, in the same two scenarios of RN in government.

France's public finances are likely to come under more strain no matter the outcome of a snap parliamentary election.

Rating agency S&P Global, which recently downgraded France, said in early June that policies advocated by the far-right National Rally could affect the country's rating.

The European Commission said two weeks ago that France, Italy and five other countries should be disciplined for running budget deficits over EU limits.

German government bond yields ticked up after German inflation data, which affected expectations about the European Central Bank easing cycle.

Markets slightly scaled back their bets on future rate cuts and priced in around 60 bps of ECB monetary easing this year, implying an additional 25 bps cut and a 40% chance of a third move, from 60% before the German figures.

Germany's 10-year bond yield, the benchmark for the euro area, rose 9 bps to 2.58%.

Inflation data from France, Italy and Spain released last week were broadly in line with market forecasts. (Reporting by Stefano Rebaudo, Editing by Anil D'Silva)