Every five years, millions of Europeans across the continent go to the polls to elect their national members of the European parliament. This May we’ll be doing so again, in what could be a watershed election for rightwing populists.

Although radical right parties won pretty big in the past two European elections, their influence within the various umbrella groups that make up the European parliament’s power blocs remained limited. This year, most rightwing populist parties may make only modest seat gains. But they also have the opportunity to create, for the first time, a serious rival to the centrist political groups that until now have dominated the EU’s governing body.

Today, the right wing’s most powerful group, and the third largest in the European parliament, is the European Conservatives and Reformists (ECR), a Eurosceptic coalition dominated by British Conservatives. Assuming Brexit goes ahead, the ECR will lose its dominant member, and so will a separate rightwing group led by Ukip. As a result, the populist right is wide open to new leaders and possibly new organisations.

On Wednesday, in Warsaw, Matteo Salvini, Italy’s interior minister and the leader of its rightwing Northern League (LN), met with Jarosław Kaczyński, the head of Poland’s Law and Justice (PiS) party and the true power behind the Polish government. They were there allegedly to discuss the formation of a new political group within the European parliament.

Although Kaczyński is anti-Russian and Salvini is one of Putin’s biggest cheerleaders, both men share a deep distrust of the EU, an intense dislike of (especially Muslim) immigrants, and a strong support for traditional Catholic values. They also share a need to protect their countries from EU pressure. Poland is facing sanctions for its attacks on liberal democracy, while Italy has been criticised by Brussels for its fiscal and immigration policies.

Salvini is currently a member of the radical right Europe of Nations and Freedom (ENF), the smallest group in the European parliament, which is dominated by Marine Le Pen’s renamed National Rally. As the dominant politician in one of the EU’s largest member states, Salvini has politically outgrown this group.

If he can bring together the ENF’s main parties with the ECR, in which Kaczyński will become the major player, this radical right ECR-plus could end up rivalling the centre-left Socialists and Democrats (S&D), which is currently the second largest group in parliament, but is set for massive electoral losses in May. Moreover, the parties that would make up the ECR-plus would have positions in the governments of many EU member states and would even have prime ministers – including Giuseppe Conti from Italy and Mateusz Morawiecki from Poland – in the European council.

Of course, it’s not a given that the Kaczyński-Salvini alliance will materialise. In the run-up to the 2014 European election, the big story on the radical right was the new alliance of Le Pen and the Netherlands’ Geert Wilders, two electorally successful but politically marginalised politicians. The media eagerly and uncritically repeated grandiose claims that they were going to create a big political group that would “wreck the EU from inside”. In the end, it took them a year to make a small group, which remained irrelevant throughout the whole legislative period.

That said, there is no doubt that radical right parties will be more prominent in the new European parliament. We are likely to see big gains from Alternative for Germany (AfD) and Vox in Spain, which will both become medium-sized parties in big countries, which means a lot of new seats in the European parliament.

Moreover, radical right parties will be even more mainstreamed than before. As the largest right-wing groups, including ECR, drift farther and farther to the right, collaboration between the mainstream and radical conservatives, both within and between umbrella groups, will become the norm in the next parliament, particularly on prime issues like immigration and security.

Will radical right parties succeed in fundamentally transforming the EU? Probably not. But they could block the reforms the EU desperately needs in order to address not just fundamental internal challenges – like Brexit and slow economic growth – but also an increasingly hostile world, dominated by two leaders, Putin and Trump, intent on weakening the EU further. The greatest irony of all, of course, is that they’ll be doing all this damage in Brussels – right in the heart of the European Union they are so intent on destroying.

• Cas Mudde is a Guardian US columnist and the author of Populism: A Very Short Introduction and The Far Right in America