I am asking this question as I remember reading somewhere that there was knowledge among the people living in one of those cities or the cities in vicinity which accurately traces back 6500 years ago. This made me wonder if the people living in this cities, say after 2000 BCE, knew how old their cities were? Do we have any records for this?
They thought they were much older than they could have been. The very beginning of the Sumerian King List has the very first king of the very first royal city in Sumer reigning for nearly 30,000 years:
After the kingship descended from heaven, the kingship was in Eridu.
In Eridu, Alulim became king; he ruled for 28,800 years.
Alalgar ruled for 36,000 years.
Two kings; they ruled for 64800 years.
This is of course, as we know, impossible. 64,800 was likely chosen for any sort of reason other than historical.
That said, you must be misremembering information about "accurately knowledge." There might be some memories that had been passed down along the ages, but no records existed before ~3300 BCE, because writing was only invented around that time. Eridu was around at least a whole millennium before writing even existed, so there wouldn't be any records of its foundation.
So without records and oral tradition, I would say no, the Sumerians didn't know how old their cities were.