You can make an offer at any level you'd like, but the new lender will always use the lesser of the purchase price or appraised value in order to determine your loan-to-value (LTV). As far as financing goes, the LTV is the critical piece of the puzzle, not the purchase price relative to the appraisal.
If you have any questions, let me know.
It the house Apprasises for $300,000
and you want to offer $290,000
the Bank just wants to know that the house is worth AT LEAST as much as the LOAN.
If you wanted to Offer, $310,000;
you would have to come up with the $10,000 difference.
Good luck and may God bless