President Lech Kaczynski and his wife Maria were among the 96 who perished when their government aircraft crashed while coming into land at Smolensk airport in western Russia over six years ago.
The exhumations are part of a new investigation into the crash initiated by Law and Justice, Poland’s governing party.
The cause of the crash has become the subject of a bitter political divide in Poland.
The current government rejects official Polish and Russian reports that concluded a combination of factors, including pilot error and bad weather, caused the disaster.
Jaroslaw Kaczynski, the twin of the late president and leader of Law and Justice, has suggested his brother was in fact the victim of an assassination plot possibly involving Russia and Donald Tusk, who was Polish prime minister at the time of the crash.
Mr Tusk, who is now president of the European Council, is a bitter political enemy of Mr Kaczynski.
Antoni Macierewicz, Poland’s defence minister, has argued the aircraft broke up in mid-air as a result of an explosion, possibly a bomb.
Both the Kremlin and Mr Tusk have denied any involvement in the disaster, although Russia’s refusal to hand back the shattered remains of the aircraft to Poland has fuelled conspiracy theories in the minds of some Poles.
Scientists will check the remains of the presidential couple to see if they were correctly identified, to determine the cause of death and to see if they carry any traces of an explosive material. In total some 83 exhumations are expected.
Mr Kaczynski’s critics have accused him of playing politics with a national tragedy, and exploiting the disaster to smear his political enemies.
Critics also point out that Law and Justice has so far failed to produce any significant evidence of foul play, despite repeatedly raising the subject since it came to power over a year ago.