HONG KONG—Lenovo Group Ltd.'s talks to buy part of International Business Machines Corp.'s server business show the Chinese personal-computer giant's next ambition: challenging U.S. rivals in the lucrative market for products and services aimed at corporate clients.

Lenovo is still a minor player in enterprise servers, even though the company is now the world's largest PC maker after it overtook Hewlett-Packard Co. and Dell Inc. A deal with IBM could turn Lenovo into a competitive player in the roughly $50 billion server...