* Greece to get 10.3 bln euros new cash, 7.5 bln of it in June

* IMF willing to join bailout as euro zone offer debt relief terms

* Dijsselbloem hails "major breakthrough", new confidence at table

* Optimism that vicious circle broken - Greece's Tsakalotos

By Jan Strupczewski, Francesco Guarascio and Alastair Macdonald

BRUSSELS, May 25 (Reuters) - The euro zone gave Greece its firmest offer yet of debt relief in what finance ministers called a breakthrough deal that won a commitment from the IMF finally to return to taking part in the bailout for Athens.

After talks that lasted into the small hours of Wednesday, the Eurogroup ministers gave a nod to releasing 10.3 billion euros in new funds for Greece in recognition of painful fiscal reforms pushed through by Prime Minister Alexis Tsipras's leftist-led coalition, subject to some final technical tweaks.

But a bigger step forward was a deal by which the euro zone agreed to offer Athens debt relief in 2018 if that is necessary to meet agreed criteria on its payments burden. That was enough to secure an agreement from the International Monetary Fund to again join the euro zone in funding the bailout of Greece.

"We achieved a major breakthrough on Greece which enables us to enter a new phase in the Greek financial assistance programme," Eurogroup President Jeroen Dijsselbloem, the Dutch finance minister, told a news conference. "This is stretching what I thought would have been possible not so long ago."

Acknowledging the "political capital" European ministers invested to reach the deal -- a nod to strong German objections to debt relief -- Dijsselbloem called it a "new phase" in a six-year drama to stabilise Greece's finances that has taken the 16-year-old euro zone to the brink of break-up.

Mutual trust was returning to the talks, he said, nearly a year after Tsipras's rejection of austerity measures pushed Athens close to be pushed out of the euro.

GREEK OPTIMISM

"I think there is some ground for optimism that this can be the beginning of turning Greece's vicious circle of recession-measures-recession into one where investors have a clear runway to invest in Greece," Tsipras's finance minister, Euclid Tsakalotos, told reporters as he left the Brussels meeting.

The IMF has long insisted on the European governments taking a hit to relieve Athens of some of its debt in order to make its public finances more sustainable. The refusal of Germany and others to do that had led to months of wrangling with the IMF in which Athens had been something of a spectator in negotiations.

While the Europeans did not offer immediate debt relief, or make an unconditional promise of reducing the payments Greece must make to them, they did spell out criteria for it. Athens' gross financing needs show be kept below 15 percent of GDP in the medium term and below 20 percent beyond that.