Enhanced Background Checks Act of 2021 (Sec. 2) This bill revises background check requirements applicable to proposed firearm transfers from a federal firearms licensee (e.g., a licensed gun dealer) to an unlicensed person. Specifically, it increases the amount of time, from 3 business days to a minimum of 10 business days, that a federal firearms licensee must wait to receive a completed background check prior to transferring a firearm to an unlicensed person. (This type of transaction is often referred to as a default proceed transaction.) If a submitted background check remains incomplete after 10 business days, then the prospective purchaser may submit a petition for a final firearms eligibility determination. If an additional 10 days elapse without a final determination, then the federal firearms licensee may transfer the firearm to the prospective purchaser. (Sec. 3) The Government Accountability Office must report on the extent to which the changes have prevented firearms transfers to prohibited persons. (Sec. 4) The Federal Bureau of Investigation must report on the number of petitions it receives for final federal firearms determinations. (Sec. 5) The Department of Justice, in consultation with the National Resource Center on Domestic Violence and Firearms, must report on further amendments to the background check process that would likely reduce the risk of death or great bodily harm to victims of domestic violence, domestic abuse, dating partner violence, sexual assault, and stalking.