* Renzi to be asked to form government Monday - party source
* Would be third unelected premier in a row
* Many Italians fed up with sclerotic political elite
* Renzi must strike deal with centre-right party on policy
* Caretaker economy minister warns turnaround takes time
By Steve Scherer
ROME, Feb 16 (Reuters) - Centre-left leader Matteo Renzi, expected to form Italy's next government this week, must overhaul one of the euro zone's most troubled economies without an electoral mandate and while awkwardly sharing power with a centre-right rival.
Enrico Letta resigned as prime minister on Friday after his Democratic Party (PD) forced him to make way for Renzi, 39, who is promising radical reforms to the euro zone's third-biggest economy and a government than can survive until 2018.
President Giorgio Napolitano is likely to ask the slick-talking mayor of Florence, who became PD leader just two months ago and would be Italy's youngest ever prime minister, to form a government on Monday, a PD source told Reuters.
That would make Renzi the third premier in a row picked by the president and not by popular vote in a country where a long entrenched political elite resistant to reform has become widely unpopular over systemic corruption and mismanagement.
"Renzi committed an original sin, which is that he will become prime minister without an election," said Giovanni Orsina, deputy head of Rome's Luiss School of Government. "Now in order to make that original sin forgotten, he needs to govern very effectively."
After getting a mandate from the president, Renzi will have to strike a deal with the small New Centre Right (NCD) party, whose support the PD needs to command a parliamentary majority.
The party, which split from ex-premier Silvio Berlusconi last year, said it wants to see a written programme that puts a clear centre-right stamp on tax, jobs and family policies before backing Renzi.
"We are decisive. If we say no to the government, it will not be born," Alfano said at a party rally on Sunday.
Backroom horse-trading for key posts in Renzi's cabinet by the NCD and other small allies is in full swing, Italian media said on Sunday, and once completed, Renzi must swear in his team and seek confidence votes in both houses of parliament.
Then the government will take the helm of an economy that grew by a meagre 0.1 percent in the fourth quarter of last year, the first sign of improving business activity seen since the country entered its worst post-war recession in mid-2011.
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In an interview with SkyTG24 television on Sunday, caretaker Economy Minister Fabrizio Saccomanni reflected on the difficulties a new government will face.
Up until two weeks ago, Renzi had refused the idea of taking power without first winning an election. But his mood shifted when Italy's main business lobby and its biggest labour union publicly abandoned Letta and called for more speed on reforms.
"In Italy there is growing impatience because everyone wants to see a rapid turnaround in growth and in job creation, especially for the young," Saccomanni said.
"These are necessities that I fully agree with and that I fought for, but one must understand that Italy's economy is like a giant oil tanker that cannot turn on a dime."
Gross domestic product has shrunk by about 7 percent in the last five years and industrial output has fallen by 25 percent. Hundreds of thousands of companies have gone out of business and joblessness has risen to levels not seen since the 1970s.
Further complicating any government's attempt to stimulate growth is the enormous public debt - at 2 trillion euros ($2.74 trillion) it is more than 130 percent of total annual economic output.
Renzi has pledged a "Jobs Act" with tax cuts for employers and a new contract scheme to promote hiring. He has said that European Union deficit rules should not trump the need to revive growth, but the previous two unelected premiers had little luck in making the major reforms they had promised.
When Mario Monti replaced Berlusconi during the 2011 euro zone debt crisis, he calmed markets with significant cuts to spending, above all on pensions.
But such austerity worsened the recession, and he failed to ease rigid rules on hiring and firing.
Letta took over a right-left government after last year's deadlocked election and satisfied EU officials by keeping a tight rein on spending, but made no major economic reforms.
In what may turn out to be an omen, the PD was disputing a tight race for the regional governorship of the island of Sardinia on Sunday, with Berlusconi's Forza Italia incumbent, Ugo Cappellacci, running head-to-head against the PD challenger Francesco Pigliaru, according to polls.
It is the first electoral test the PD has encountered since Renzi won a party primary in December, and he would no doubt like for it to go better than the soccer match between his Fiorentina and Inter Milan on Saturday.
Renzi attended the game, shaking hands and mixing with fans at the Florence stadium after working on his government team all day in the mayor's office. Inter, which had not won an away game since November, defeated Fiorentina 2-1.