I just answered a question from someone who has an FHA loan and is having a hard time getting his offer accepted. Here's my answer to his question. Hope this helps turn on your thinking cap.
Some agents may not tell you, but I have seen in the confidential remarks of the listings, NO FHA loans. Not that they won't look at it, but rather, it's because FHA loans (as well as VA) must conform to very strict guidelines before loans are approved.
For REOs, where many of the properties are trashed and/or stripped, these may not pass the FHA guidelines. For example, a functional kitchen requires a stove/oven. If the REO (or even short sale) is stripped of this, you can't get your loan approved.
Also, $20K may not be enough in a competitive situation. Many of the short sale and REO listings are listed "artificially" low to get buyers excited enough to write offers. The problem is, there is a lot of excitement at certain budget ranges. Recently, my buyers were contemplating writing an offer on an REO --- but by the time I got hold of the REO agent, he had already received 98 offers!
Another competitor is the all-cash buyer. Many of these investors have formed a buyer consortia where they buy properties all cash, no contingencies. Even if the offers are lower than others, the lenders may decide to accept the cash offers because they are, in their view, a SURE THING.
TYPE OF LOAN
Has your lender determined that you will only qualify for an REO loan? Or is there a possibility you can get a conventional loan? Knowing what your options are, you may be able to position yourself more competitively.
CHANGE YOUR SEARCH PARAMETERS
If you are preapproved for $300-400K, have you tried to lower your expectations, and search for properties under $300K? Then you'd have a little more wiggle room to submit an offer higher than the $20K you've established.