WASHINGTON—Thousands of State Department and U.S. Agency for International Development employees indicated in a survey they are worried about the future of their agencies, with some expressing particular concern about lack of support from the Trump administration and Secretary of State Rex Tillerson.

The findings are in a report for Mr. Tillerson compiled by a consulting firm as he embarks on an effort to reorganize the State Department amid steep budget cuts. The Wall Street Journal reviewed a copy of the report, which is...