The country has been hurtling towards a political crisis since the opposition resigned en masse in February, launching weekly protests calling for the resignation of the Socialist Party's Prime Minister Edi Rama.
Now the battle is intensifying after Meta tried to scrap the June 30 vote that the opposition was boycotting. He said on Saturday that the poll would no longer be "free nor democratic" without their participation.
But on Monday, Socialist MPs responded saying parliament would "adopt a resolution on the impeachment of president". That will go before MPs on Thursday.
Prime Minister Edi Rama had already dismissed the president's move, vowing on Saturday to press ahead with the elections, which put the pair at loggerheads.
But it is not clear that the Socialists can remove the president.
A final decision on the matter would need to be made by the Constitutional Court, which is currently out of action.
In recent months all but one judge on the bench either resigned or was dismissed as part of a reform process aimed at rooting out corruption.
"This procedure does not deprive us of the right to start the process of deposing the president," Socialist party spokeswoman Elisa Spiropali told reporters.
Speaking to journalist on Monday President Meta said that cancelling the election was necessary to "avoid a civil war and destabilisation of the country.
"Parties should promptly find an agreement to break the deadlock," he added.
The president has no official party affiliation, but before assuming office in 2017 Meta led the then junior ruling Socialist Movement for Integration party, which is now in the opposition.
The opposition coalition, which draws from right-wing and centre-left parties, accuse Rama of crime and corruption links.
Their protests have at times turned violent, with demonstrators hurling Molotov cocktails and firecrackers as police respond with tear gas and hoses.
The unrest is seen as especially ill-timed for Rama, who is hoping this year to get a greenlight from the European Union to start accession talks to join the bloc.