Renegotiating an offer

Technically, until a contract is signed by both parties, no one party is legally bound to the other. Oral contracts concerning real estate are not enforceable. Rather, non-legal collateral considerations come to bear in the situation you outline. It is delicate to re-open negotiations and it should be very well-considered with a game plan in place. You must be ready to lose the deal completely. Whereas initial dealings were likely marked by a presumption of "good faith" by each party, the party abandoning a prior "hand-shake" agreement will no doubt incite some degree of animosity making the re-negotiating process more difficult on an interpersonal level. Without details, it is difficult to say more.