(Reuters) -Two U.S. citizens and one Russian were among 20 people detained at protests in Tbilisi while Georgian lawmakers were debating a "foreign agents" bill that has sparked a political crisis, Russia's TASS state news agency reported on Monday.

Georgia's opposition had called on opponents on Sunday of the bill to stage an all-night protest outside parliament to prevent lawmakers from entering on Monday.

Georgian Prime Minister Irakli Kobakhidze vowed on Sunday to push ahead with the law, after opponents of the bill rallied in one of the largest protests seen since independence from the Soviet Union in 1991.

TASS reported, citing witnesses, that police started to push the protesters away from the service entrances of the parliament building early on Monday, leading to some scuffles.

The "foreign agents" bill requires organisations receiving more than 20% of their funding from abroad to register as agents of foreign influence or face fines. The ruling party says it is needed to enhance the transparency of NGO funding and protect the country from outside interference.

Western countries and Georgia's opposition denounce it as authoritarian and Russian-inspired. Critics liken it to Russia's 2012 "foreign agent" law, which has been used to hound critics of Vladimir Putin's Kremlin.

The dispute over the bill has come to be seen as key to whether Georgia, which has had traditionally warm relations with the West, continues its push for European Union and NATO membership, or instead builds ties with Russia.

The EU, which granted Georgia candidate status in December, has repeatedly said the bill could jeopardise Tbilisi's further integration with the bloc.

Russia's RIA news agency said the judicial committee of Georgia's parliament approved the third reading of the bill on Monday, clearing the way for the full parliament to vote this week on completing its passage through the legislature.

(Reporting by Reuters in Moscow and Lidia Kelly in Melbourne; Editing by Mark Trevelyan)