Of couse you can offer less.
However, this strategy has consequences.
A typical foreclosure is priced below market value.
This attracts lots of buyers.
The process is structured to get multiple offers.
If you don't come in strong and fast....
....it dissappears into the 'black hole' of losing foreclosure offers.
Some banks WILL systemicly send out a 'highest and best' counter to all received offers to advise buyers it time to get serious if this is the home you want.
Be very aware, there is more to consider than price. Many homes sell to buyers who do not offer the highest price.
It has been my expereince, buyers asking this question seldom purchase a home until they realize they are not assessing the competive nature of the REO market place properly. When they have properly assessed this market they ask, "How can I get this house?" not "How low can I go?" Even those with the proper mindset find it very competive.
If you don't change your thinking, you will soon be submitting a question similar to:
"Why are all my offers rejected? What must I do to win in a Foreclosue bid?"