With that many offers it will be hard to predict your chances... but 24k over list seems like a good starting point!
I see you are in the Bay area where prices are considerably higher than in the Sacramento region so $22K might make sense. Might...
If you absolutely MUST have that property, there is a great chance of you getting the property IF your offer is all cash and can close the fastest. With REO properties, the lender is looking strictly to see who can net them the most. Cash does not always matter. Be prepared to have a number that you will cap out at. It's always best to offer your best and highest out the gate in the event the seller chooses not to counter.
My best to you!
Keisha Mathews, REALTOR®
CDPE®, HRC®, HAFA® Certified
"The Short Sale Lady"
Century 21 Landmark Network
(916) 370-1803 cell/direct
(916) 405-3886 fax
The Bank will probably come back asking for your "BEST and FINAL" offer.
I would presume that you know what the Market Value of the house is; that you've done a CMA:
Be aware that if the M.V.is $550,000 and you bid $600 because you LOVE it, you will still lose, because the Appraisal will not cover the $600: The Bank will only lend a maximum of 80% of the Appraised Value, therefore $440,000, and you would have to come up with $160,000 (the down payment would overlap some).
You have to decide how much you love this house and act unemotionally; there are other houses out there.
Good luck and may God bless
and Happy Mother's Day!