The IRS gives guidance only for what the buyer is allowed to do with regard to real estate taxes. Publication 530, 2011 edition, says the following
(where I have
added emphasis to point out the similarity to the Illinois real estate
You bought your home on May 3, 2011. The property tax year in your area is the calendar year. The taxes for the previous year are assessed on January 2 and are due on May 31 and November 30. Under state law, the taxes become a lien on May 31. You agreed to pay all taxes due after the date of sale. The taxes due in 2011 for 2010 were $1,375. The taxes due in 2012 for 2011 will be $1,425.
You cannot deduct any of the taxes paid in 2011 because they relate to the 2010 property tax year and you did not own the home until 2011. Instead, you add the $1,375 to the cost (basis) of your home.
You owned the home in 2011 for 243 days (May 3 to December 31), so you can take a tax deduction on your 2012 return of $949 [(243 ÷ 365) × $1,425] paid in 2012 for 2011. You add the remaining $476 ($1,425 − $949) of taxes paid in 2012 to the cost (basis) of your home.
Extrapolating from this, I would say that for a sale in 2012, on his
2012 tax return (due in three months time), the seller (OP)
can deduct all the real estate tax for 2011 (assessed in 2012, due
in 2012, and paid in 2012) regardless of whether the buyer
or seller made the actual payment(s) during 2012.
The taxes for the January 2012 - September 2012 period (which will be
assessed in early 2013 and payable during 2013) and for which the
seller has paid the buyer
at the time of closing reduce the capital gains made on the house
by the seller and increase the basis for the buyer.
Neither the buyer nor the seller can deduct the real estate taxes for this period
(January 2012 - September 2012) on their 2013 income tax returns.