Census from 28AD done by Augustus puts population of Italy at 4 mil. The problem is we don't know if it only included male citizens or all the population. If it's the former then adding women, kids and slaves makes it closer to 20 mil.
The best surveys from antiquity come from china. At around year 1AD population of china reached about 60 million people, area 6,000 km2.
Let's extrapolate these numbers on Roman Empire.
First we have to notice that area occupied by China is and was the most fertile land on Earth. Lot's of moisture and rice crops twice a year. 60 mil figure doesn't look like anomaly, because census done a hundred years later confirmed 50 mil people.
http://en.wikipedia.org/wiki/Census
Surely there are always doubts. How accurate the bureaucrats were getting to every part of huge country, how efficient was the system, or how honest were government administrates. For the lack of any proofs of otherwise, let's assume that these surveys were pretty accurate statistically.
Now, Roman Empire covered approximately similar area in it's farthest extent. The nature of land mas was not so fertile though. Half of empire that's desert, semi-desert or mountains, plus crops only once a year. These conditions could probably produce food of 1/3rd of what fertile china could. It means 60/3=20mil, population of Roman Empire.
If we take figure of 4 mil from Augustus census, and assume that this figure includes all the people in area of todays Italy, then it makes sense. It makes sense that one fifth of people of empire lived in it's center, which includes one million in center of a center, Rome.
We have to keep in mind that in agricultural societies ratio of city dwellers to farmers is 1 to 9. It takes 9 farmers to feed their families plus to make extra food for one city family. One could say that if Rome was one million strong plus the rest of Italy's cities and towns, let's say 2 million all together. According to ratio 1 to 9, we would end up with 20 million people in Italy.
Rome at it's hight was a special place though. People from around empire flocked there to live. Rome was rich and could afford that. Shipments with grain and animals were coming to Rome from all over empire every day.
At the time Lombards arrived, empire was in ruin. Rome sucked few times, people are very poor, and grain shipments stopped coming.
In this case I put population, in area of today's Italy, by year 600AD at around 2 million.
Off course this figure could fluctuate drastically in decades. Two or Three years of drought and we are down to one million. Couple of decades of bumper crops and population doubled to 4.