It depends on if the Feds suffered a loss and/or if the conviction resulted in the prospective borrower being placed upon one of several Federal lists.
Part of an FHA Underwriter's function is to verify that a prospective borrower for an FHA insured mortgage is not barred from participating in FHA insurance.
There are several background checks made on borrowers; FHA comes with four of its own in addition to the standard checks.
Here re the FHA background checks:
The first two are open to the public:
All parties to the transaction - buyer(s), seller(s), Realtor(s), appraiser, loan originator, settlement agent - are run through both of these background checks.
The second two are limited to authorized users:
In addition to the Excluded Parties List and LImited Denial of Participation, the prospective borrower must also pass USA PARIOT and CAIVRS (Credit Alert Interactive Voice Response System). The former is generally run with a Trimerge Credit Report; the latter is run when the lender obtains an FHA Case Number.
All lenders are supposed to do the following - FHA, VA, USRDA, Conventional:
The borrower will also be run against the OFAC database (Office of Foreign Asset Control, which also manitains the "No Fly List"). Many lenders use software that bundles OFAC checks with loan compliance verification from companies like Wiz Sentinel.
Additionally, the property appraisal will be checked by either software or a second appraiser. Generally these are designed to grade the appraiser on risk of the value estimate, but some also run checks on the property's history (eg past reports of toxic waste, mold, or radon gas) and the location (close to a gas stattion, dry cleaners, or other potential source of toxicity).