The crisis talks come after months of violent demonstrations with protesters calling for Mr Martelly to stand down.
His opponents accuse him of stalling the elections which were due in 2011.
The crisis comes as Haiti prepares to commemorate Monday's fifth anniversary of a devastating earthquake which destroyed most of the county's infrastructure and left hundreds of thousands of people living in temporary camps.
The country has been slow to recover with around 80,000 people still living in squalid tent camps and only 67% having access to latrines.
Two weeks ago, Mr Martelly announced he had reached a deal with the Congress to solve the crisis.
Under the agreement, Haiti's deputies would keep their seats until 24 April and senators until 9 September.
The Senate is yet to vote on the proposal.
Prime Minister Laurent Lamothe resigned on 14 December and was replaced by Evans Paul, a former radio journalist.
But the move was not enough to end the street protests which continued, with renewed calls for Mr Martelly's resignation.
Mid-term Senate elections had been originally due in May 2012, while the municipal poll is three years behind schedule.
They were postponed again on 26 October - the day they were due to be held - because of the stalemate between the government and a group of opposition senators over an electoral law.