According to preliminary reports, Vareikis was due to fly to Rome on Thursday morning for a discussion organized by the International Catholic Legislators Network (ICLN).
News outlets said the MP was not allowed to board his aircraft, as he appeared to be heavily intoxicated.
Meanwhile Greta Babarskaitė, communications coordinator for Vilnius airport, told BNS that no incidents involving Vareikis were reported.
Police officers who observed Vareikis at the airport also said he did not commit any offenses and was therefore peacefully escorted to a taxi.
Conservative leader Andrius Kubilius said he would meet with Vareikis to discuss whether his alcohol addiction keeps him from performing his duties as a member of parliament.
Meanwhile Razma, first deputy leader of the party's parliamentary group, said Vareikis should give up his mandate, noting that "the political group and the party, as well as the entire Seimas, will have to make strict decisions", in case he fails to tender resignation himself.
Earlier this year, Vareikis was charged in another incident he was involved in while being drunk.
According to data provided by the police, the MP committed a physical offense against his wife in the couple's home in the outskirts of Vilnius on May 4.
