If you have a 620 FICO score, you have to work with banks that allow scores down to 620. Major banks do require 640 or higher, however FHA guidelines allow scores down to 580 with 3.5% down and down to 500 if the buyer puts 10% down. So yes, there are lenders who can do low FICO score FHA loans. We used to do 3.5% down payment loans down to 530 until FHA established their minimum FICO amounts a few months ago.
We offer a 1/2% down payment FHA program with a 580 FICO. If you qualify, you get 3% down payment assistance and must contribute at least 0.5% of the purchase price. Other costs to purchase include an appraisal ~$450, home inspection ~$400 and you will have to pre pay your annual homeowners insurance premium and the lender may require a property tax installment to be paid (if closing near April or December).