Banks prefer cash offers, And they will often accept a lower cash offer over a higher offer that involves a mortgage. Do your homework. Work with an Realtor to determine what is a realistic estimate of the home's value after it's fixed up. Then I would reduce that amount by 10 to 15% to allow some room for error. Estimate the cost of repairs. Get real. Don't use low ball numbers.
Are you going to live in the home if your offer is accepted???? or are you looking to flip the property????? If you're going to flip the home, you need to factor a profit into your calculations.
Now you can determine your maximum offer. In most cases you probably want to your initial offer to be less that your max. But not always. Why?? Because sometimes you don't get a second chance to bump it up. Remember, your offer is competing against other offers. So while your waiting for a counter offer and hoping one of your incrementally increased offers will be accepted, ideally below your maximum, someone else's offer may be accepted.
You need to think about how strongly do you want the home.
Your agent will submit your offer for you. Usually in a few days, it varies, it could be more or less, the bank will accept, reject, or counter-offer.
Sometimes the bank will have the broker handling the foreclosure/REO notify all those who have submitted an offer that they (the bank) have received multiply offers. They ask all those who have submitted an offer to submit their highest and best offer which must be received by a designated date and time.
The bank will choose (but they are not obligated) the offer they consider best from their standpoint. For example they may choose $125,000 cash over a $140,000 offer which is comprised of $20,000 cash down payment and $120,000 mortgage.