Yes, if you are eligible for the visa waiver scheme. The US rules say 90 days, but there is an 'exception' clause for UK and some other passports that means that if your passport is valid for less, you will be admitted until it expires. Do NOT overstay this period.
why dont you renew your passport now ? they will tag on the extra 3 months, and as it is not a popular holiday time at the moment you should get your passport back fairly quickly, especially if you use the Post Office Fast track service, I got mine bak in 6 days.