Repayment of Signing Bonus and Relocation Expenses

We have an employee that was paid a signing bonus and relocation and signed the agreement that he would have to pay both back if he left before 1 year. He is leaving after 4 months and wants to only repay the net amount of the moving expenses and bonus actually received (i.e., moving expenses/bonus amount received after income withholding and FICA deducted). He says the company can then recoup such withholding and FICA by amending our quarterly tax report (form 941) and claiming a refund for over-payment of these taxes. He also says his 2018 W-2 should not include this income or the related withholdings, as it should be essentially treated as never being paid upon his repayment. What are the rules for repayment? Should he pay the gross or as he wants to, the net amounts? Is there a publication we can show as reference for the rules?