ATHENS (Reuters) - Parties supporting and opposing Greece's international bailout are running neck-and-neck ahead of next month's repeat national election, which could determine the country's future in the euro zone, three opinion polls showed on Friday.

Greece was forced to call the new June 17 vote after an election on May 6 left parliament divided evenly between groups of parties that support and oppose the austerity conditions attached to a 130 billion euro bailout agreed with the European Union and International Monetary Fund in March.

The failure of the two parties that dominated Greece for decades - conservative New Democracy and Socialist PASOK - to win a pro-bailout majority, and the success of the anti-bailout radical leftist SYRIZA party which came second, have sent shockwaves through Europe.

Of three polls published on Friday, two put anti-bailout SYRIZA ahead but two of the polls showed gaps of less than 2.2 percentage points between SYRIZA and New Democracy.

Greek election rules give the party that comes first an automatic bonus of 50 seats in the 300-seat parliament, so that even a slim advantage could play a decisive role in determining which party forms the next government.

A poll by VPRC for the Kontra TV station showed SYRIZA was ahead with 28.5 percent of the votes, and the pro-bailout New Democracy second with 26 percent.

Another poll earlier on Friday by RASS for the Metro newspaper showed New Democracy was ahead with 23.6 percent of SYRIZA with 21.4 percent, while a third poll by Metron for TV station Antenna showed a narrower gap with 27.2 percent for SYRIZA and 27 percent for New Democracy.

The Socialist PASOK, which also backs the international bailout keeping Greece afloat, remained steadily in third place in all three polls.

New Democracy and PASOK will be trying to scare voters who punished them in the last election into returning, arguing that electing SYRIZA leader Alexis Tsipras would mean the end of Greece's euro zone membership.

Polls since the May 6 election have shown SYRIZA, New Democracy and PASOK improving their performance at the expense of smaller parties.

An overwhelming majority of more than 75 percent of Greeks want to stay in the euro, but two thirds oppose the bailout, which came with harsh salary, pension and job cuts.

EU leaders have warned Greece that if it renounces the bailout they will pull the plug on funding, leading to rapid bankruptcy and an exit from the single currency.

(Reporting by Karolina Tagaris, editing by Tim Pearce)