The Lithuanian was nominated by the group of the European People's Party representing right-wing forces, however, he failed to secure support of all of the assembly. Zingeris' supporters believe this could be due to his rigid stance on Russia.
Zingeris' competitor for the post, Stella Kyriakides, refused to withdraw from the race even after being defeated by Zingeris in the group election.
Although a member of the group of the European People's Party, Kyriakides is officially running as an independent candidate.
Two votes at the assembly in Strasbourg on Tuesday did not provide any of the candidates with the necessary majority. Zingeris was behind the Cypriot both times, with the third vote scheduled for Tuesday afternoon. It could be the decisive ballot, as a simple majority will then be enough to be appointed to the post of PACE president.
Kestutis Masiulis, a member of the Lithuanian delegation at PACE, the vote was secret, however, debates within the group make it clear that Zingeris does not have the support of the parliamentarians who want Russian deputies back in the assembly. Russia's membership was suspended following the annexation of Crimea.
"The countries that want Russia back in the organization support the representative of Cyprus. Meanwhile, Emanuelis is backed by the countries that believe that Russia's return is impossible until it withdraws from seized Donbass territories. Emanuelis made it clear that, if appointed as president, he would do everything to have Russia out until it fulfills the requirements," Masiulis told BNS on Tuesday.
In his words, the presidency position currently belongs to the European People's Party, therefore, rejection of the group's official candidate is destroying established rules.
"When the turn comes for the Socialists, we will nominate a candidate. We do not plan on following the rules, if they don't support our candidate," said Masiulis.
The Parliamentary Assembly of the Council of Europe unites parliamentarians from 47 states, mainly working on human rights issues and making decisions of advisory nature.
Spanish MP Pedro Agramunt stepped down from the assembly's president earlier this month amid criticism of visiting Syria with Russian politicians.