Mr Aliot, vice president of the RN, suggested far-right, eurosceptic groups must focus on creating “a union of sovereignists in the European Parliament” to “influence European policy”.
He told the French news channel BFMTV on Monday: “The bloc’s sovereignists, if they prove able to join forces, could lead the second biggest group in the European parliament.”
Mr Aliot argued a sovereignist bloc in the EU chamber would be a “huge” deal, before adding: “If we have the opportunity to become the second or third biggest group in the new parliament, we must seize it.”
The RN’s newly elected MEPs have already started negotiations to form a new political alliance, according to Mr Aliot, who added that Brexit Party leader Nigel Farage, a staunch eurosceptic, had been invited to take part in the talks in an effort to “build a more important group”.
French far-right leader Marine Le Pen’s RN is currently a member of the Europe of Nations and Freedom (ENF) group, which also includes members of Italian prime minister Matteo Salvini’s far-right League party.
Various populist, anti-Brussels and right-wing parties made gains overall in the European elections, winning more than 150 seats and finishing in first place in France, Italy and Britain.
But while most agree that member states are shackled by EU rules and have surrendered too much sovereignty to Brussels; they remain bitterly divided on other key issues, casting doubt over their chances of turning what may be about a quarter of parliament’s seats into a united force.
Mr Salvini and Ms Le Pen have been trying to form a “supergroup” of like-minded, anti-immigration parties, but their progress has been hindered by conflicting national interests.
The ruling hard-right nationalist party in Poland, the PiS, for example, has so far refused to align with Mrs Le Pen’s RN partly because of her pro-Russian views.
In an interview with Polska Times.pl published last week, Poland’s prime minister Mateusz Morawiecki said the PiS was open to forming a parliamentary alliance with Italy’s League and Spain’s far-right Vox party, but not with France’s RN.
He said: “We are flexible, but we rule out an alliance with any faction that would have France’s Rassemblement national as a member.”
Hungary’s far-right Prime Minister Viktor Orban has also ruled out joining forces with the RN.
Mr Farage has also rejected any tie-up with Ms Le Pen in the past over the history of anti-Semitism in the French far-right.
The EU elections also usher in weeks and possibly months of hard bargaining over who will run the bloc’s institutions.
The 28 EU national leaders, including outgoing Prime Minister Theresa May, will meet over dinner in Brussels on Tuesday to discuss the succession to Commission President Jean-Claude Juncker and other key jobs, including the head of the European Central Bank and the European Council, in light of the election results.