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 the country'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 vote, but the final outcome will depend on days of alliance-building before next week's run-off vote.

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

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

The spread between French and German 10-year sovereign bond yields - a gauge for the risk premium investors demand to hold French bonds – tightened to 73 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.

"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.

Other yield gaps tightened, with Italy and Greece down 8 bps each at 149 and 113, respectively, while Portugal fell 6 bps to 66 and Spain 1 bp to 85.50. Austria and Belgium's spreads tightened by about 4 bps.

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 dealmaking made by its rivals over the coming days.

Citi analysts recently said the yield gap would tighten to 70-75 bps if RN leads the government, implements 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 trigger an increase in the debt risk premium in Italy, the bloc's most vulnerable country, as credit rating agencies see an expansionary debt path. The gap could rise to 140 bps and 155 bps, respectively.

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 could affect expectations about the European Central Bank easing cycle.

Figures from France, Italy and Spain were broadly in line with market expectations.

German 10-year bond yield, the benchmark for the euro area, rose 7 bps to 2.56%.

Markets priced in around 60 bps of ECB monetary this year , implying an additional 25 bps rate cut and a 40% chance of a third move in 2024.

(Reporting by Stefano Rebaudo, Editing by Anil D'Silva)