WARSAW (Reuters) - Poland's Prime Minister Donald Tusk opened the door on Tuesday to a popular vote on joining the euro, with surveys showing most of the country's 38 million population would vote no.

Tusk had previously shied away from a referendum on the single currency, which the Poles are obliged to adopt under EU membership, but has had to agree to a formal vote in exchange for necessary changes to the country's constitution.

"I would be in favor of reaching an agreement to change the constitution, where there would be a referendum about joining (the euro zone)," Tusk told a news conference.

Recent polls suggest a large majority of those in what is by far the European Union's largest eastern member state, oppose joining the euro zone, put off by four years of turmoil that has crushed economies in Greece, Spain and others.

Tusk's government holds a slim majority in the parliament and would need the support of the rightist Law and Justice (PiS) party to reach the two-thirds in the lower house required to change the constitution to swap zlotys for euros.

PiS has previously said it would oppose the euro adoption without a referendum.

Officials have previously said they would focus on meeting euro entry criteria and would only set an official date after the 2015 general election.

Analysts polled by Reuters expect the European Union's largest eastern economy to adopt the euro in 2019.

(Reporting by Pawel Sobczak; editing by Patrick Graham)