Sheffield manager Paul Heckingbottom called the incident "assault" and pledged that the perpetrator "will be dealt with."
According to the British media, Nottinghamshire police have launched an investigation into the attack.
Nottingham Forest released a statement saying they were "appalled" by the footage.
"The club will work with the authorities to locate the individual in order they are held to account for their actions, which will include a life ban from Nottingham Forest," it read.
"The club would also like to apologise to Billy personally and to Sheffield United."
Sheffield United manager Paul Heckingbottom told Sky Sports: "There's a prison sentence there for sure. We're seeing things thrown on the pitch and things that are thrown on putting players' health and safety in danger and nothing's ever done about it. There will be something done about that. We've seen what's happened, we know what's happened.
"How can you come to a place of work minding your own business and be assaulted in that fashion? It's just bang out of order and it's everything that's wrong. Too many times we hear people condemn it, but nothing's done about it. We're the ones that suffered.
"It's assault. We've seen one of our players attacked. He's shuck up, bleeding, angry. It'll be dealt with.
"It's upsetting for a lot of reasons - I don't want to say too much more about it. He'll be dealt with."
A two-time European champion, Forest was last in the top flight in 1999. The team can return there by beating Huddersfield at Wembley Stadium on May 29 in one of the most lucrative games in world soccer because of the riches on offer in the Premier League through its broadcasting deals and prize money.
Huddersfield was promoted to the Premier League in 2017 after 45 years in the lower league, before getting relegated in 2019.
