Butkevičius said so after meeting with Seimas opposition groups on Tuesday morning. He claims to have received "information from certain services" about alleged ties of teachers' unions with Russia.
The prime minister insists that government negotiators had almost reached agreement with teachers' representatives on all points on Friday and were going to seal the deal on Monday, but the trade unions presented additional demands after the weekend.
"The position has changed and it is due to some external influences. I am not afraid to say that Russia's influence on certain unions is great, I will not go into details," Butkevičius told reporters.
He refused to specify whether Russia was funding trade unions in Lithuania.
About 230 primary and secondary schools across the country, one-tenth of the total, started a strike on Monday, demanding higher wages and reforms in the education system.
The Ministry of Education and Science has offered to set aside €5 million to give a 1.5% salary boost this year, but said that any additional funding would require a vote in parliament to revise the national budget.
Prime Minister Butkevičius said he could not give more details about his allegations, saying the data was classified.
He only said that Lithuanian teachers' unions had ties with those in Russia and regularly went to conferences in Moscow and Saint Petersburg. "Who could deny that afterwards it's their [Russia's] proposals and demands that get implemented here in Lithuania," Butkevičius said.
Parliament Speaker Loreta Graužinienė said she had not received any reports from Lithuanian special services about ties between teachers' unions and the government in Moscow.
Commenting on the prime minister's allegations, Seimas opposition leader Andrius Kubilius said he was "shocked".
"I suspect that the prime minister is simply making irresponsible statements which are in keeping with the general line of the prime minister and the ruling coalition to dismiss any criticism - from the opposition, the media, or trade unions - as enemy conspiracy," Kubilius said.
Eligijus Masiulis, leader of the opposition Liberal Movement, was even more strong-worded.
"I am concerned about the prime minister's state of health. I feel like the prime minister could use help from either a priest or a psychiatrist, since we keep hearing more and more conspiracy theories from him," Masiulis said.
He added that Butkevičius would need to give evidence to support his statements.
Union representatives have responded to Butkevičius' allegations by calling his statements "politically blind" and demanding that he give names of individuals who are allegedly influenced by Moscow.
"I think that the prime minister shows his political immaturity when he talks about developments in the education system. Such political blindness is surprising," Eugenijus Jesinas, chairman of the Lithuanian Education Institutions' Union, told BNS.
He said that at least members of his union had not participated in events organized by Russia or Russian trade unions.
"We would like that individuals linked to Russia be named," Jesinas added.
Audrius Jurgelevičius, chairman of the Lithuania Education Union, said that Butkevičius' statements were slander.
"In his war against fellow citizens, the Lithuanian prime minister has pulled out the 'Russians are attacking' cliché. It is an argument that sane self-respecting people would not dignify with comments," Jurgelevičius said, adding that his union might sue Butkevičius for slander.
He also said that the prime minister's statements about an agreement between government and teachers' representatives having been reached last Friday were inaccurate.
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