* Greens set for best showing yet in Euro election
* Role seen in "progressive coalition"
* But Greens will still be one of the smallest groups
By Alissa de Carbonnel and Matthew Green
BUDAPEST/BRISTOL, May 22 (Reuters) - An ebb in support for mainstream parties is raising hopes among Europe's Greens that they could act as kingmakers in the next European Parliament, with growing concerns over climate change likely to hand them their strongest showing yet.
With the centre-left and centre-right bloc set to lose their joint majority in elections across the European Union this week, the Greens hope to increase their influence as parties jostle to create new alliances in the next EU legislature.
There will be limits to what they can do, but the Greens want to push for faster cuts in greenhouse gas emissions, as well as for influence in the areas of trade, tax and the rule of law in any negotiations to form a bloc to counter gains on the eurosceptic far-right.
"They can do their numbers," Bas Eickhout, a Green lawmaker from the Netherlands and one of the group's two lead candidates, said of the conservative and socialists predicted to lose seats.
"We are not going to make the same mistake as other parties to say: 'Okay, we team up with you and we make some minor changes at the fringes,'" Eickhout told Reuters.
The socialist group aims to include Greens and Liberals in a broad centre-left coalition, their leader in the parliament, Udo Bullmann, told Reuters.
French President Emmanuel Macron, in an interview with the Belgian newspaper Le Soir, also suggested that there could be a role for the Greens in such a "progressive coalition".
Officials in the European People's Party, the biggest political family in the chamber, have also signalled readiness to coordinate policy with the Greens.
Despite their expected gains, the Greens are projected to remain one of the smallest political groups, with 57 seats in the 751-seat legislature, up from 52 at present - limiting their influence over big decisions.
"They could have leverage and – if they play it smartly – they could get something out of that, but let's not overstate it," said Janis Emmanouilidis, a political analyst at the Brussels-based European Policy Centre.
As negotiators urge countries worldwide to boost commitments under the 2015 Paris Agreement to curb greenhouse gas emissions, the EU's stance as one of the key brokers of the deal will be crucial in determining how much of a contribution it can make towards averting the most catastrophic climate change scenarios.
Winning seats in the European Parliament, one of the bloc's three lawmaking bodies, offers lawmakers a say on decisions over EU rules in areas such as car pollution standards, financing for pipeline projects and targets on renewables and energy-saving.
Nevertheless, interests in the energy, farming, auto and other sectors mean Greens will have to fight to translate climate activism into concrete policy changes.
Support for Green parties has surged in what was once the continent's industrial heartland, where young people have taken to the streets to demand politicians break with a legacy of dependence on coal, oil and gas.
Students inspired by 16-year-old Swedish climate activist Greta Thunberg are planning to a join a school strike in dozens of countries on Friday, while environmental activists Extinction Rebellion shut down parts of London last month, forcing the British parliament to declare a symbolic 'climate emergency'.
Although a groundswell of concern has prompted many parties to step up their green rhetoric on the campaign trail, the pan-European alliance of more than 30 national parties who field lawmakers to the Green group in the European Parliament hopes to reap the biggest rewards.
"It's moved beyond the green niche. People are ready for really bold action," Molly Scott Cato, a British Green lawmaker, said at a rally of about 100 students at the University of Bristol in southwest England.
With traditional loyalties in flux, the Greens have seen their biggest gains in Germany, where they doubled their vote in Bavaria last year. The party is polling high in Belgium, the Netherlands and Denmark, and Greens are in government in Luxembourg and Sweden.
But support is limited in the former Soviet bloc nations and in southern Europe, where concerns over migration and the economy dwarf climate fears.
The Greens hope to win their first seats in Poland, Slovenia and Hungary. "At a certain point it's then even more important to get your foot in the door in a new country," said Eickhout, stumping in Budapest for one of the opposition candidates.
But in marked contrast to big climate protests that have taken place in Western Europe this year, only a handful of teens attended a "Fridays for Future" rally in central Budapest, where Eickhout turned up for a meet and greet.
"Not many are aware of global warming here," said Ambrus Somosi, 24, a university student, who said he would likely vote not for the Green candidate but another opposition party.
Green optimism has also been tempered by the advance of right-wing populists - liable to return more candidates who question the need for emissions cuts, or dispute climate science.
And projected gains from 6 currently held by British European Parliament members in the Green alliance to more than 10 will be short-lived if the country leaves the EU as planned.
As the parliament shrinks after Brexit, 27 of Britain's 73 seats will be redistributed to candidates elected onto a reserve list in other countries. The Green group stands to lose on those.
With British lawmakers playing an outsize role pushing the climate agenda, some fear Britain's departure from the EU could leave more room for states such as Poland to dilute the bloc's ambitions.
The last European Parliament pushed for higher goals on a raft of new regulations and backed a call for the EU to achieve climate neutrality by 2050.
For many young first-time voters, that goal is paramount.
"Corporations are looking out for themselves and not the larger population," said Dani Salvalaggio, 21, an English literature student taking time out from exams to watch Cato's speech. "Other things are more important than profit."
(Reporting by Matthew Green in Bristol, Alissa de Carbonnel in Budapest and Brussels Additional reporting by Francesco Guarascio in Brussels Editing by Giles Elgood)
Our Standards: The Thomson Reuters Trust Principles.