The red terra cotta building next to Lord & Taylor at 15 W. 38 St. that has served as owner Hudson’s Bay offices has been net-leased by Jay Suites.

The Big Apple-based co-working space provider is the third-largest in the city, and its No. 1 rival, WeWork, bought the Lord & Taylor flagship store last fall and is expected to use the building’s upper floors for co-working space.

The 12-story building leased by Jay Suites has been connected with Lord & Taylor through a skybridge that runs east-west over a tiny building between them. Permits were approved in early March and the skybridge is currently being removed.

The 17-year sublease for the 90,000 square-foot building, a likely cost-cutting move by Hudson’s Bay Co., provides Jay Suites with its eighth Manhattan location.

Founded in 2009 by brothers Juda Srour, the chief executive, and Jack Srour, the president, Jay Suites concentrates on private luxury suites. It intends to spend roughly $10 million to upgrade the 1909 structure.

The brothers will open a Jay Café coffee shop in a portion of the retail space as well as a Jay Conference facility in 22,000 square feet on the first to third floors. Jay Suites will use the remaining floors while a new roof deck also is planned.

A former WeWork and Cushman & Wakefield executive, Sean Black of BLACKre, represented Jay Suites, while Laura Pomerantz and Maria Travlos of Cushman & Wakefield repped Hudson’s Bay.

The building is owned by the Rosen Group and was initially leased by Associated Dry Goods in 1984 under a 40-year ground lease that was extended by Hudson’s Bay for another decade.

Associated was a predecessor to both May and Federated Dept. stores, which sold Lord & Taylor to the Canadian-based retailer in 2006.