No. You will need to claim the repayment in a different way.
It is possible for the employer to reduce your reported wages, but they don't have to do it that way. For example, if you earned $10,000 and repaid $20,000, they could issue a W-2 with $1 of wages, but that would leave you with a $10,000 repayment to deal with. Or, they could issue a W-2 for the full amount of your wages ($10,000) and then you have to deal with a $20,000 repayment.
There are 3 parts to getting your taxes back on the repayment.
#1. Federal tax return. You can deal with this in one of two ways, either as a special itemized deduction, or a special IRC 1341 claim of right credit. If you already itemize your deductions, and your income is about the same in both years, then both methods will give about the same result. Also, you can only use the claim of right credit if you had a reasonable belief at the time the bonus was paid that you had an unrestricted right to the money. If audited, this could be tricky, since you probably signed a contract that said that you would repay the money if you failed to meet certain conditions. That could be viewed as not having a reasonable belief that you had an unrestricted right to the money, especially if you knew when you accepted the bonus that you might not meet the conditions. This is a legal issue I can't advise on, you might want to seek professional help. You can always use the special itemized deduction for the repayment, no matter what the reason for your departure was.
If you think you are entitled to use the claim of right credit, then you can decide to use either the deduction or credit, whichever saves you the most tax. To use the special deduction, enter it under "less common deductions" at the bottom of the deductions and credits page. Click through the options that don't apply to you and eventually you will see "repayment of wages from a prior year."
To use the credit, you must figure out on your own, how much less your taxes would have been in 2019 (or whatever year the bonus was paid) if you had not received the bonus. That is the amount of your credit. to enter the credit, you must use Turbotax installed on your own computer from a CD or download; switch to "Forms" mode, go to Schedule 3, and manually enter the amount on line 12d, and write "IRC 1341" on the space next to the amount. If you use Turbotax Online, you can't make manual adjustments; however, if you use Turbotax Live (the most expensive version) your Live CPA assistant might be able to make the entry for you.
#2 State income tax. You will need to research your state taxes to see how to enter a repayment of wages or claim of right credit. As there are more than 40 states with income tax, I'm not an expert on how to enter the state tax adjustment but there probably is one.
#3. Social security and medicare tax. To get a refund of the social security and medicare tax that was withheld on the bonus, you will need to file form 843. This is separate from your tax return and not supported by Turbotax. You will need to submit copies of paperwork related to the bonus and repayment (contract, demand letter, etc.) and you will need a letter from the employer stating that they will not be adjusting or repaying your social security and medicare tax. (Of course, if your wages with the employer were more than the social security wage base -- $137,700 -- for the year you received the bonus, then reducing your wages by the amount of the bonus might not have reduced the social security tax you would have paid, so you would only be entitled to a refund of the medicare tax.)
