What I might try to do is find out why the board turned down the 3 previous applicants. Then, if you're able to, make your offer as strong as possible--so that the owner will have a sense that your offer might be accepted by the board. The stronger that element is, the lower your offer to the owner can be.
I can't give you a magic number. I don't know what other units have been going for, or anything else. The hold-up at this point sounds like it's the co-op board, not the owner. If that's the case, consider making an offer for what the person owes on the unit (or what the person owes plus, say, 8% to cover real estate agent fees and other miscellaneous expenses).
I know that the others were turned down for a variety of reasons. I believe someone lied on their application and the other had young adult children in the picture. I was told I should not have a problem. This board tends to like people of my age in in my profession (I'm a teacher)