It greatly depends on which city you are talking about, as well as the period. But overall, it would be somewhat unlikely, especially if the merchant in question is not a part of Latin Christendom.

Concretely: in the Early Middle Ages, trade was an affair of high specialization, undertaken by people who were usually under the patronage of a figure of authority. Earlier in medieval historiography, it was believed that merchants were relatively pariah people who traveled constantly transporting small high value items by their own means, but later research revealed that commerce was already of a much greater magnitude than that.

In later developments, the profession of merchants started to become less dependant on patronship of noblemen, and new forms of cooperation arose, most notably, temporary, one-off associations between merchants to undertake specific enterprises and voyages. In these scenarios, there was usually a part that undertook the financial cost and risks, and another one that took charge of the journey itself. One tremendously good place to study these dynamics is Amalfi, in Italy, where the economy grew largely dependant on the activities of merchants.

This frame of commercial activity didn't allow for the establishment of any significant presence of companies in foreign territories, in that they would rather just transport goods from one place to another, maintaining a solid base in their homeland.

In the Late Middle Ages, this started to change somewhat, with the establishment of Genoese and Venecian colonies in the East Mediterranean. However, after the Ottomans started to take hold of the region in the 14th and 15th centuries, trade started to become problematic for these Italian states that relied on commerce to sustain their economy, and particularly for Venice. Their response had two main premises: they somewhat retreated into the Italian Peninsula, seeking an agricultural expansion in the north-east of it, and to the south, across the Adriatic, and at the same time, they reinforced their commercial presence in the Iberian Peninsula, particularly in the east coast of it, and some of the south. By this time, great companies of bankers (like the Peruzzi, the Bardi and the Medici companies), had already risen to prominence and fallen from it. The new frame of commercial activity had become a more elaborate development of that established in the High Middle Ages: wealthy merchants that are no longer under noble patronage establish relations of inter-dependance among themselves, sharing risks and benefits, now relying on a newly founded (and fragile) system of complex credit, the founding pillar of later capitalist banking.

All of these dynamics of trade gave little scope to the permanent presence of lonely merchants in foreign cities. In any event, you would have large congregations of Genoese and Venecian merchants established in colonies, or in districts within other cities, but this was mostly limited to certain places in the East Mediterranean and the Iberian Peninsula. There were also significant numbers of foreign merchants in Flanders and other centers of commerce in the region, including Italian, French and English merchants, but as you may see, all of these international trade relations were somewhat restricted to the sphere of Latin Christendom.

Sources and further reading:

Kenneth Bartlett, The Italian Renaissance.

Edwin Hunt, The Medieval Super-Companies: A Study of the Peruzzi Company of Florence

Guidi Bruscoli, "John Cabot and his Italian financiers"

Steven Epstein, An Economic and Social History of Later Medieval Europe, 1000–1500.

Georges Duby, The Early Growth of the European Economy: Warriors and Peasants from the Seventh to the Twelfth Century

Daniel Power et al., The Central Middle Ages. 950-1320.

Steven Runciman, The Fall of Constantinople. 1453. (This one is highly entertaining, and deals with the Venetian colony in Pera).