According to tradition, the founding of Rome by the legendary figures Romulus and Remus occurred in the year "753 BC". Of course, ancient Roman scholars would not have referenced that date using the (supposed) birth date of Jesus Christ as a starting point.
I have never seen that date in any other form than as a "BC" year. How do ancient Roman sources specify the founding date, and how did modern historians determine it to be equivalent to 753 BC?
I am not sure about legends (are there any specifying Rome's foundation year? I suspect it might be in the form of "X years since the sack of Troy"). But since your question also asked about "ancient Roman scholars" and "ancient Roman sources"...
During most of the Roman Republic, years were named based on who had been elected to the consulship for that year. Since Rome was a monarchy, obviously the consular year system did not reach far enough back to the city's foundation.
When Quintus Fabius Pictor began working on historical annals of Rome, he created a new dating method as an alternative to Consular years. Instead, it tracked the number of years since the foundation of Rome: ab urbe condita, lit. "from the foundation of the city". Not all ancient writers agreed on the same foundation year, however. The popular 753 B.C. cited in the question was calculated by Varro Reatinus. Using his system, the year 753 B.C. could then be given as "1 ab urbe condita" or "1 a.u.c.". Varro's epoch became the accepted standard for giving dates in a.u.c. after the mid 1st century or so.
Another system was the Greek Olympiad dating. The first Olympics was thought to have been held in 776 B.C.; this, and subsequent Olympics, provided a conveniently universal (to Greeks) epoch for dating. From about the 3rd century B.C. onward, some Greek writers began giving dates on that basis. For instance, the foundation of Rome has been given as "the first year of the seventh Olympiad" by Dionysius - although that would be 752 B.C., so 753 B.C. would be something like "the fourth year of the sixth Olympiad".
Of course the ancient Romans didn't date the foundation of Rome to 753 BC.
My answer to this question:
Correspondence between the modern and the antique calendar1
Says that ancient scholars calculated several different years for the foundation of Rome beside the 753 BC that later became the standard date. The years they calculated were expressed in Greek Olympiad dating and other dating systems used in the ancient world. Ancient historians, like modern ones, often converted dates from one dating system to another one.
So ancient writers estimated that Rome had been founded in 814 BC, 748/7 BC, 719/8 BC, 752/3 BC, 751 BC, and 752 BC, as well as the 753 BC that became the official date.
The modern system of dating ancient events by years BC only became common in the 17th century less than 400 years ago.
