The politicians discussed amendments to a 2020 presidential election law already passed by the Sejm, the lower house of parliament.
"The talks make sense, the meetings make sense and (...) this new round of round table talks has shown that talks can lead to agreement, that the (ruling) coalition can find common ground with the opposition," Gawkowski said at a press briefing following the talks.
"After this meeting, one thing is certain: we want the presidential elections to take place as quickly as possible and the date of these elections is most probably June 28, 2020," said Gawkowski, quoting a declaration made by politicians from the ruling Law and Justice (PiS) party.
Gawkowski said the electoral legislation, which is now being processed by the Senate, should return to the Sejm with amendments next week "to conclude the process so that (Sejm) Speaker Elzbieta Witek can announce the (date of the) elections."
PiS's Przemyslaw Czarnek said that by August 7 a new president must be elected and the election rubberstamped as having been valid. The party's Marek Suski said all caucuses agreed that the new president should be sworn in by August 6, when the current president's term of office ends.
"This meeting ended in success as we've managed to convince the whole opposition that the elections should take place so that Poland can have stable authorities," Suski said.
Opposition parties want to make a large number of amendments to the presidential vote bill prepared by PiS, arguing the legislation process was not thorough enough as the bill was rushed through the Sejm in a single day.
Senate committees are set to discuss the bill on Thursday.
However, the main opposition group, the Civic Coalition (KO), was less upbeat about the talks. "I believe that PiS is sticking to a single date - no concessions, which is absurd from the point of view of fairness and the principles of a democratic state," PAP was told by KO MP Robert Kropiwnicki.
The round table talks are expected to continue next week, Gawkowski said.
The Sejm passed the election bill on May 12, and the Senate, which received the legislation a day later, has 30 days to process it. The draft law assumes that the voting will take place in polling stations, but citizens will also be allowed to cast their votes by post if they register for this latter option well ahead of the ballot.
A previous 2020 presidential vote bill, prepared for the now-cancelled May 10 vote, envisaged only postal voting amid the coronavirus epidemic but raised concerns over fairness and adherence to election standards.
At the last moment, the May 10 election was postponed by Jaroslaw Kaczynski, leader of PiS, and Jaroslaw Gowin, who heads PiS's junior coalition partner, Agreement. (PAP)