The aircraft was carrying the high-level delegation to a ceremony to mark the 70th anniversary of the Katyn massacre, where 22,000 Polish officers were killed by Soviet forces in World War II.
Attendance in doubt
The attendance of almost 80 foreign delegations, including Barack Obama, the US president, and Dmitry Medvedev, the Russian president, was in doubt after Poland joined other European nations and closed its airspace on Friday due to the spread of volcanic ash coming from a volcano in Iceland.
Kaczynski's family insisted the funeral go ahead as planned and the government said there had been no cancellations by foreign leaders so far.
Jonah Hull, Al Jazeera's correspondent in Warsaw, said concerns about the volcanic ash had already led to some foreign delegations cancelling their trip.
"The Indians, the South Koreans and the Japanese won't be taking the journey," Hull said.
"Barack Obama's trip is still on and we're told Nicholas Sarkozy, the French president, will make a decision at the last minute on whether to come.
"We've heard the Slovakians and Estonians said they will travel here by road if necessary."
After the mass in Pilsudski Square on Saturday, the closed coffins of Kaczynski and his wife Maria, who have been lying in state in the presidential palace since Tuesday, will be taken to nearby St John's Cathedral.
Kazimierz Nycz, Warsaw's archbishop, will lead a service there followed by an overnight vigil.
The bodies will be moved to the southern city of Krakow on Sunday for the funeral in the cathedral of the hilltop Wawel castle, where Poland's past kings and national heroes already lie.
The family's decision to bury Kaczynski, a divisive political figure in life, at Wawel castle sparked protests during the week but they have died down.
Meanwhile, Russian and Polish investigators are continuing to investigate the cause of the crash.
Officials said that they suspected pilot error following the first analysis of the "black box" recorders of the Russian-made Tupolev Tu-154, with some suggesting the pilot ignored advice from air traffic controllers to divert to another airport because of the fog.
Elections to replace the president are expected to take place on June 20, but no formal announcement has been made.