Midtown’s 41-story office tower at 590 Madison Avenue, long known as the IBM Building, is on sale at roughly $1.1 billion.

A Manhattan office skyscraper on Madison Avenue is up for sale, in one of the first major tests of the market for trophy office building sales this year.

On a full-block site, a rarity in high-density Manhattan, an enormous new corporate headquarters has risen. For its bulk, it is almost obsessively unshowy, though its oversize windows, framed in ...