* League leader says party has right to govern
* Anti-establishment 5-Star also claims leadership role
* Long period of political stalemate forecast
* Italian assets under pressure (Adds Renzi resigning, Macron comment, Salvini seeing Berlusconi)
ROME, March 5 (Reuters) - Two anti-establishment leaders made early plays to govern Italy on Monday, sending ripples across the euro zone after voters relegated mainstream parties to the sidelines in delivering a hung parliament.
With the bloc's third-largest economy seemingly facing prolonged political instability, the anti-immigrant League claimed the right to rule after its centre-right alliance won the largest bloc of votes.
"We have the right and duty to govern," League leader Matteo Salvini told a news conference, saying investors should have no fear, as the prospect of a eurosceptic-led administration promising to ramp up spending hit shares, bonds and the euro.
Minutes later, the head of the biggest single party, the anti-establishment 5-Star Movement, said it was ready to take on a responsible leadership role.
"We're open to talk to all the political forces," 31-year-old Luigi Di Maio said in a statement. "We feel the responsibility to give Italy a government (as) ... a political force that represents the entire nation."
With the vote count almost complete, none of the three main factions had enough seats to govern alone. President Sergio Mattarella is not expected to open formal coalition talks until early April.
In Brussels, a European Commission spokesman said it was confident a stable administration could be formed. "And in the meantime Italy has a government with whom we are working closely," he added.
Salvini criticised both the euro and European Union-wide restrictions on national budgets. "The euro was, is and remains a mistake," he said, although he added that a referendum over Italy's continued participation in the single currency was "unthinkable".
The rightist alliance that also includes former prime minister Silvio Berlusconi's Forza Italia (Go Italy!) was on course for around 37 percent of the vote -- but for the first time the League emerged as the senior partner.
The role reversal marks a bitter personal defeat for the billionaire media magnate and his party, which took more moderate positions on the euro and immigration while the far-right League campaigned on a fiercely anti-migrant ticket.
Berlusconi, who has not been seen in public since the vote, met Salvini on Monday in his villa near Milan.
Afterwards, Forza Italia said it was open to welcoming in other parties to create a centre-right government.
Before the encounter, Salvini told reporters that, while not interested in a broad "minestrone" coalition, the League would be willing to talk to all parties. The party's economics chief Claudio Borghi said their first contact would be with 5-Star.
Anti-establishment parties have been on the rise across Europe since the 2008 financial crisis.
In Italy, where the economy is 6 percent smaller than a decade ago and unemployment is stuck near 11 percent, Sunday's biggest loser was former prime minister Matteo Renzi's Democratic Party (PD) that has ruled since 2013.
The party secured less than 20 percent of the vote, its worst ever showing, prompting Renzi to resign as leader. Nonetheless, he vowed the PD would not join any new coalition.
"The Italian people have asked us to be in opposition and that is where we will go," Renzi told reporters, adding: "We will never form a government with anti-system forces."
Despite overseeing a modest recovery, the Democratic Party's centre-left coalition trailed on 22 percent, also paying the price for widespread anger over an influx of more than 600,000 migrants over the past four years.
The European Union has insisted that Italy take charge of the newcomers, letting hardly any of them move to other EU states. This stance has helped erode Italian faith in Europe and French President Emmanuel Macron said on Monday the election result was a reminder of problems tied to migration.
"Italy has, it's undeniable, suffered for months and months under the pressure of migration," Macron said in Paris.
Prolonged political stalemate could make heavily indebted Italy the focus of market concern in Europe. Investors dumped Italian government bonds on Monday and the euro was under pressure, while the Milan bourse dipped 0.5 percent.
During two months of election campaigning, party leaders repeatedly ruled out post-ballot tie-ups and if no cross-party pacts can be found, a swift return to the polls is possible.
Formed in 2009, 5-Star has fed off public fury over institutional corruption and economic hardship. Some have questioned whether other parties would be able to work with it.
"Di Maio wins, Italy ungovernable," was the front page headline in La Stampa newspaper.
Having ditched earlier ambitions to quit the euro, 5-Star's flagship proposal in the election campaign was a minimum monthly income of up to 780 euros ($963) for the poor, helping it draw massive support in the underdeveloped south.
By contrast, Salvini and Berlusconi -- heading for almost 18 percent and 14 percent respectively -- appeared to have dominated the wealthier north, with support for the centre-left largely concentrated into a narrow stretch of central Italy.
"It's what we expected, political deadlock. A government is needed," Rome resident Giuseppe Bruni told Reuters. "We are in the hands of the president and the willingness of political parties to put a government together."
($1 = 0.8097 euros)
(Additional reporting by Gavin Jones, Francesca Piscioneri, Giselda Vagnoni and Isla Binnie Writing by John Stonestreet and Crispian Balmer Editing by Steve Scherer and Catherine Evans)