By Crispian Balmer

ROME (Reuters) - When a handful of European leaders met Barack Obama in Berlin this month to say their goodbyes, Italian Prime Minister Matteo Renzi informed the group that he may well lose power before the U.S. president.

While Obama leaves office on Jan. 20, Renzi has promised to resign if he does not win a Dec. 4 referendum on constitutional reform, opening the way for renewed political instability in the eurozone's third largest economy.

"I have no desire to hang around if I lose," Renzi told the gathering, according to a diplomatic source who was at the low-key Nov. 18 meeting.

Opinion polls now predict Renzi's defeat, in what would be the third big anti-establishment revolt by voters this year in a major Western country, following Britain's unexpected vote to leave the European Union and the U.S. election of Donald Trump.

Pressure is mounting on Renzi to drop his threat and instead agree to remain in power to deal with the fallout from a 'No' vote, including the risk of a fullblown banking crisis.

Obama himself said in October that Renzi should "hang around for a while no matter what" and a number of businessmen and senior government officials contacted by Reuters said they feared the worst if the prime minister abandoned his post.

"My personal opinion is that Renzi should stay," Industry Minister Carlo Calenda said in an interview on Friday. "What needs to be considered ... is what is good for the country."

Three center-left politicians who are in regular contact with Renzi told Reuters that he would honor his word and immediately resign if he is beaten, worried that failure to do so would do irreparable damage to his political image.

IMPULSIVE

The Italian president could appeal to Renzi's sense of responsibility and ask him to seek a new mandate from parliament. His response might depend on the size of any defeat, with one advisor saying the 41-year-old premier could quit politics altogether if he suffers a huge snub next Sunday.

"He is young and impulsive," said the official, who declined to be named. "If the result is terrible, he might decide to call it a day and do something else with his life."

The referendum proposes constitutional reforms to strengthen the lower house of parliament and reduce the authority of the upper house Senate. Regions would lose some decision-making powers to bolster central government.

Renzi says the project is necessary to make Italy, which has had 63 governments since 1948, governable enough to enact reforms needed to revive its moribund economy. Opponents say it would reduce democratic checks and balances.

In the first week in December, Italy's third largest bank Monte dei Paschi di Siena has to launch a 5 billion euro ($5.3 billion) cash call. Investors are not expected to jump in if political instability prevails, meaning the state will almost certainly have to intervene swiftly to stave off collapse.

Although business leaders support Renzi's reforms, they have kept quiet during the referendum campaign, fearing they would prove the kiss-of-death in an era of anti-establishment angst.

But they are increasingly alarmed by the prospect of heavily indebted Italy drifting once more into political paralysis.

"I am afraid that if he loses the referendum (Renzi) really will give up," Ferruccio Ferragamo, chairman of fashion house Salvatore Ferragamo, told Reuters. "Italy can react to anything, but this really would be a step back."

NO SAFETY CAR

Opinion polls cannot be legally published in the final two weeks of campaigning, but the last 40 surveys released before the Nov. 18 cut-off showed the 'No' camp ahead by up to 11 percentage points.

A source in Renzi's Democratic Party (PD) said on Friday that private polls suggested this gap had closed to five points with a quarter of voters still undecided, meaning victory was still possible.

Initially the plan was backed by 70 percent of Italians, but when an over-confident Renzi said at the end of 2015 he would resign if defeated, opposition parties turned the referendum into a de facto ballot on his 2-1/2 years in office.

His record is mixed. Despite many reforms, Italy is set to have the third lowest growth in the 28-nation European Union in 2016 and the second lowest next year, according to EU forecasts. Unemployment is stuck above 11 percent and wages are stagnant.

A loss would provide more evidence of voter fury in Europe ahead of elections in France and Germany next year. And Renzi's exit could benefit populist ex-comedian Beppe Grillo, who wants to ditch the euro currency and whose Five Star movement won more than a quarter of the vote in Italy's last general election in 2013.

Renzi acknowledges that personalizing the vote was a mistake and in August he changed tack, refusing to discuss his future while on the campaign trail. But with polls showing no sign of recovery, the prime minister has doubled down on his original threat over the past two weeks.

"I'm not willing to take part in old-style political games. Either we change or I have no role to play," Renzi said this month -- a line he regularly repeats at the frenzied round of rallies and media interviews he is undertaking before Dec. 4.

If he quits, it is not immediately clear what would happen next. The straightforward answer would be for Italy to hold national elections a year ahead of schedule.

But Renzi was so confident of victory in the referendum that he introduced a new electoral law in 2015 just for the lower house, believing the Senate would no longer be in play. To avoid using different electoral systems for the two houses, parliament would have to devise a new election law, which could take much of 2017.

President Sergio Mattarella, the supreme arbiter of Italian politics, could ask Renzi to oversee this reform as head of a so-called "single-purpose government", but the prime minister's allies say he would never agree to such a limited mandate.

"Renzi can't serve as a safety car," PD lawmaker Matteo Richetti told Reuters, referring to the car that takes to the track after accidents in Formula One motor racing.

If Renzi is not prepared to be a figurehead leader, government officials contacted by Reuters said Economy Minister Pier Carlo Padoan or Senate speaker Pietro Grasso were the most likely candidates to step into the breach.

Padoan, a former official at the International Monetary Fund, would be seen as particularly market-friendly in what are likely to be turbulent times.

But a Padoan or Grasso government could not take office without Renzi's blessing, because it will need the backing of his PD party to survive. Renzi's allies fear he will be held responsible for its actions even if he is not prime minister.

"All those who are demanding a 'No' vote should support a government that prepares the way for elections. But that won't happen. They are going to create an almighty mess and expect us to clear it up," said Richetti.

(Additional reporting by Giselda Vagnoni, Silvia Ognibene and Andreas Rinke in Berlin; editing by Peter Graff)