Hopefully, you have a realtor working for you to help you through the process.
Everything hinges on how responsive the lender is, and how good your offer package is. You may only get one shot at this, so make sure you submit your best and highest offer, and that your package is complete with your preapproval letter, copy of your initial deposit. In some cases, the lender also asks for information on the buyer's FICO score. To further expedite the process, you should state this is an AS IS offer. Sign all the disclosures if they're already provided, and submit these with your offer.
Depending on the degree of completeness, you may even be able to get an answer within 1-2 weeks(which is totally different from short sales that may take upwards of 45 days to get a response!).
Sometimes, the highest offer doesn't always get accepted. If the terms are good (like shorter or no contingencies, shorter escrow, higher initial deposit, higher down payment, etc), a lower offer may beat a higher offer.