I can't speak to ancient cities or the enormous Italian cities before the plague, but early modern London certainly had a large "vagrant" population.

Here are some estimates on the size of London in the early modern era (Wrigley, "A Simple Model of London's Importance..." Past & Present July 1967 pp. 44-70):

1600 - 200,000

1650 - 400,000

1700 - 575,000

1750 - 675,000

Around 1700, London was the largest city in western Europe, exceeding Paris by about 75,000 people. London also had a much higher proportion of the population of England than any other city in western Europe. Between 1650 and 1750, about 10% of the population of England lived in London. However, this population increast over the course of 150 years was not entirely due to “natural increase” (i.e. high birth rates). London was a bit of a death trap in the early modern period, with high rates of disease, and death rates exceeding birth rates. The reason the population of the city grew was because of incredibly high migration rates from the countryside to the city. (Wrigley & Schofield, Population History of England, 1541-1871). The poorer sort in England had a significant level of geographic mobility, so long as they could find work. And most of the migrants tried to find work in London, with varying degrees of success (Slack, “Vagrants and Vagrancy in England, 1598-1664”, Economic History Review n.s. August, 1974, pp. 360-379).

It is not possible to know the exact number of vagrants living in the city, but there are frequent references in the Acts of the Privy Council regarding the vagrant population (Examples at the bottom). There are also references to vagabonds in the records of the Old Bailey online Example. There were dozens of laws passed by parliament to deal with poor, both in the city and in the country, most of which dealt with licensing beggars, punishing “able poor” and proscribing methods of transporting poor back to their home parish (Slack, The English Poor Law, 1531-1782, appendix). For most of the early modern period, people who were unable to work (such as the lame or aged) were the responsibility of the parish they were from, but the huge population of London and limited belief in a “right” to poor relief, made enforcing the rule difficult (Hindle, On the Parish? The Micro-Politics of Poor Relief in Rural England, p. 402). The few surveys of the poor population we have indicate that there was something like 5-15% of the population in London were homeless or “vagrants”. However, these are rough estimates at best. The persistent presence in government correspondence and parliamentary legislation indicates that the poor & vagrant populations in the nation, and especially in London were a serious social problem in Early Modern England (Slack, Poverty and Policy in Tudor and Stuart England).

Additional References:

APC, 1625-1626, p. 18 – All streets to be cleaned and vagrants punished & suppressed for the funeral of James I in London.

Ibid, pp. 251-252 – Increase of vagrant population due to decrease in trade, and death from plague. Paraphrase: There being nothing more scandalous to the government of the State in the eyes of foreign nations…to see the wealthiest and principal city of the kingdom…swarm and abound with so many poor and beggarly people unrelieved or set to work…”

APC, 1628-1629, No. 1259: “We have in former times written diverse letters to [the Mayor and Aldermen of London] for the suppressing of Vagabonds and Wandering persons remaining in the city of London…”

A Venetian resident in London also referred to the large number of vagrant persons; CSPD, Venice, 1673-1675, pp. 446-448.