An FHA approved contractor is actually a DE lender (Direct Endorsement FHA Lender). You can find these DE lenders by going to hud.gov.
However, most people think that contractors who build, remodel and renovate properties can be FHA approved. However, HUD/FHA does not approve, endorse, educate or certify construction contractors. Builders can become HUD approved to construct residential properties for HUD.
However, if you are referring to the 203k loan, there is no such thing as an FHA approved contractor because HUD/FHA does not approve, endorse, educate or certify construction contractors for the 203k loan. Instead, the borrower is allowed to hire any contractor they want for the 203k loan but it is the responsibility of the lender making the loan to "approve" the contractor by verifying the contractor's credentials, work experience, licensing information, type of work performed, experience, and client references. But since lenders do not have an education system for contractors for the 203k and most do not have sufficient infrastructure to properly "approve" contractors for the 203k loan, lenders can direct these contractors to http://203kContractors.com
to become educated on the 203k. These Certified 203k Contractors will enable the lender to be in compliance with HUD's Mortgagee Letters 98-11 and 00-25.
203k Contractor Directory for the FHA 203(k) Loan