Both Roy Keane and Patrick Vieira added their condemnation and concern after Aston Villa goalkeeper Robin Olsen became the latest player to be attacked by a fan during a post-match pitch invasion.
Manchester City have apologised to Olsen and vowed to indefinitely ban the fan who struck him during Sunday's pitch invasion after City clinched the title.
Supporters have run onto the field after a number of recent high-profile Premier League and EFL matches, with a Nottingham Forest fan jailed for 24 weeks after headbutting Sheffield United's Billy Sharp.
Crystal Palace manager Vieira was involved in an altercation after Everton fans invaded the pitch on Thursday and fans streamed onto the field again after City's 3-2 comeback win against Aston Villa which secured the Premier League title.
Thousands were on the Etihad Stadium turf after wrapping up the Premier League title and Villa boss Gerrard says goalkeeper Olsen was hurt.
Asked if everyone got off the pitch OK, Gerrard said: "The answer is no. My goalkeeper was attacked. I think those questions should go to Pep and Manchester City."
City swiftly released a statement apologising to Olsen.
"Manchester City would like to sincerely apologise to Aston Villa goalkeeper Robin Olsen, who was assaulted after the final whistle at today's match when fans entered the pitch," the club said.
"The club has launched an immediate investigation and once identified, the individual responsible will be issued with an indefinite stadium ban."
Speaking on Sky Sports, Roy Keane said: "It's disgraceful, it really is. And I tell you, a player or a manager is going to be seriously injured.
"If you're mad enough to come on and punch a player, you're going to be mad enough to do something silly like stab a player. There will be something crazy.
"People have come back after Covid and forgotten how to behave themselves. Idiots, scumbags, disgrace.
"A player walking off a pitch and getting attacked. We saw it during the week a couple of times. At Forest, we saw it with Patrick Vieira. Absolutely disgraceful."
Speaking after Palace's win over Manchester United on the final day, Vieira added: "I think the Everton incident is bigger than just that incident.
"When you look at it in the Premier League, there were a lot of field invasions that can cause trouble. There are real issues here to reflect on and try to see what is the best way to manage it and to deal with it.
"You don't know what can happen. When you don't know what can happen of course you fear for anything, for the players.
"We want to be safe and the best way to do that is we need to reflect on what is the best way for those kind of things not to happen on the field. As a coach, manager, player or staff we want to be safe in our workplace.
"I think there is a big issue on the FA and Premier League's plate to deal with.
"They need the support and the club and the players to get involved. Everybody has to take responsibility and see what is the best way for those things not to happen."
Vieira revealed he would be happy to speak to the FA about his own incident but stressed England faced a country-wide problem.
"I will participate with the FA and talk about the incident and give my opinion and my view," Vieira added.
"Again, it is not about just that incident. It is about all the incidents and I think it happened too often in the last couple of weeks and we need to reflect on that and make a decision so players, staff and fans can be safe going to watch football matches."