The tenement at 700 East 134th Street in Port Morris shares the block with a stand of clapboard row houses, a metalworker and a document shredding facility. Tenants, many of whom have lived on this quiet, gritty block in the South Bronx for the better part of a decade, pay around $1,000 a month for their apartments.

Soon, they could pay much more.

In many ways, this little building exemplifies the anxiety echoed throughout the South Bronx as middle-class New Yorkers priced out of Manhattan stream across the Harlem and East Rivers in search of more affordable apartments. Tenants fear that the wave of wealthier newcomers is pushing rents up and poorer residents out.

In early January, the 21-unit building containing mostly studios sold for $4 million to a limited liability company after the previous owner filed for Chapter 11 bankruptcy. Soon after the sale, James Giddings, the head officer of the limited liability company, began telling tenants to expect rent increases of around $500 a month, according to tenants.

Mr. Giddings has spoken with tenants in the hallways, in their apartments, on the phone and via texts and emails. For years, the previous owner did not sign leases with tenants, meaning the residents essentially have only 30-day agreements. Without a lease, they are vulnerable to sudden rent hikes or losing their apartment.