By Crispian Balmer

ROME (Reuters) - Ex-Prime Minister Silvio Berlusconi has anointed Stefano Parisi, a former Internet executive and government economic adviser, as his political heir, giving him a mandate to relaunch Italy's splintered center-right.

Parisi, 59, narrowly lost a mayoral election in Milan last month, impressing friends and foes alike with an upbeat campaign that came close to knocking the center-left from power in the country's financial capital.

Looking to build on that experience, Parisi said last week he wanted to unite Italy's disparate conservatives and has now secured the blessing of Berlusconi, who asked him carry out a profound review of his own Forza Italy (Go Italy!) party.

Berlusconi, who is 79 and underwent major heart surgery in June, said in a statement on Tuesday that he wanted Parisi to "come up with a project to relaunch and renew the position of moderate Italians in the political sphere".

After months of relative calm in the often turbulent world of Italian politics, tensions are once again building ahead of a referendum on constitutional reform set for the autumn.

Centre-left Prime Minister Matteo Renzi, 41, has said he will resign if he loses the vote. With the latest opinion polls running against him, there is a risk that general elections might have to be held more than a year ahead of schedule.

Surveys show that if the conservatives unite behind a single leader they could win roughly the same support as Renzi's Democratic Party and the anti-establishment 5-Star Movement.

"If Renzi loses the referendum, then we will need a candidate for the premiership and I want it to be you," la Repubblica newspaper quoted Berlusconi as telling Parisi.

There was no immediate confirmation from Forza Italia. The party has been in disarray since Berlusconi was banned from public office following a conviction for tax fraud in 2013, but despite the internal chaos it remains a pivotal group on the fractured center-right.

Parisi is a former CEO of Internet and telecoms company Fastweb and managing director of industry lobby Confindustria. Although his only elected experience is as a city councilor in Milan, he was previously a technocratic economic adviser to both center-left and center-right prime ministers in the 1990s.

Known for his soft-spoken moderate tone, Parisi will need to develop a strategy for working with the anti-immigrant Northern League, Berlusconi's long-standing populist ally in government.

The League polls at around 13-14 percent nationwide and considerably higher in the north, but the antics of its leader, Matteo Salvini, have offended moderates. On Monday, he raised fresh hackles by comparing the woman speaker of parliament to an inflatable sex doll.

Under Berlusconi the center-right never won a national election without the support of the Northern League, and in recent public remarks Salvini has been cool toward the idea of Parisi as the new leader of the conservative bloc.

(Additional reporting by Massimiliano di Giorgio; Editing by Paul Taylor)