You can list moving expenses as an itemized deduction in the program. They are not allowed for federal taxes, but will flow to your state, and may be deductible depending on your state and if you itemize deductions on your state tax return. (NY does allow the deduction, if you itemize.) If you report all your expenses and your taxable reimbursement, Turbotax will do the proper calculation. As far as federal tax law is concerned, moving expenses are currently not deductible. You got a taxable bonus, and it doesn't matter what you spent it on.
