Apex Telemarketing began construction Tuesday of a new 34,000-square-foot facility at Sorensen Research Park to handle toll-free calls and marketing programs nationwide, spokesman J. Robert Howe said.

The new building, Apex's fourth Utah telephone facility, is scheduled for completion in summer 1989.William Francis Construction Co. has been named builder and developer of the project financed by First Security Bank of Utah.

Apex employs 600 people in three Salt Lake facilities. The new building will add 374 work stations to handle telemarketing projects for firms nationwide.

In addition to handling incoming and outgoing telemarketing calls, the 10-year-old firm does consulting, software development, shipping, data entry, credit card processing and national message services.