If genuine, the find in St Peter's parish church in Carmarthen, west Wales, would solve a 160-year-old mystery about the whereabouts of the descendants of the king and explain why the monarch donated a pipe organ to the church.
The tombs were found by archaeologists re-tiling the church's historic floor which had subsided.
They and the vicar at the church, Canon Randolph Thomas, said today they had had no idea that the bodies were buried beneath the floor.
One of the tombstones is engraved with the name Charlotte Dalton, and the date of her death 1832 which is the same as George III's granddaughter who died aged 27 in that year.
She was from his first marriage, which was kept secret.
'Beats expectations'
The Prince of Wales's wedding in 1759 to Quaker girl Hannah Lightfoot and the births of their three children, were concealed to allow him to marry Princess Charlotte of Mecklenburg-Strelitz.
The lord chancellor of the day reportedly confiscated the marriage documents and they are believed to held in the Public Record Office.
Charlotte Dalton's mother Sarah married a Carmarthen man James Dalton and Church records said she was buried in the town but did not give an exact location.
The other grave in the historic church bears the name of Charlotte Dalton's niece Margaret Prytherch, and the date of her death aged nine in 1839.
Mr Thomas said: "We had no idea that we were going to find the tombs as there is no record of them in history.
"We did expect some interesting finds but this beats all our expectations."