Professional Footballers' Association chief executive Gordon Taylor says action must be taken to prevent football fans invading pitches to confront players before a “tragedy” occurs.
Two fans were arrested for separate pitch invasions on Sunday, when Jack Grealish was punched during Aston Villa's game against Birmingham, while Chris Smalling was pushed during Manchester United's defeat to Arsenal.
A fan was also arrested following an ugly incident involving Rangers captain James Tavernier during Friday night's 1-1 draw at Hibernian.
Taylor believes the incidents are part of a worrying trend of an increase in anti-social behaviour among a minority of supporters in Britain this season.
"It was disgraceful, cowardly and criminal but we have seen an increase in such behaviour at clubs this season," Taylor told Sky Sports News.
"There has been racist abuse, there has been anti-Semitism and now there is real fear for the safety of players.
"From that point of view, we had a meeting just the other week with the Minister for Sport, Mims Davies, about engaging on this issue between the police and the stewards, which is crucial, and the responsibility of clubs.
"We need to find the right balance between educating supporters who are there, and also having sanctions.
"We have to make sure we put a lid on it. We have been down this road before and we do not want to go there again and see the game threatened."
Taylor would like to see the government take a lead in addressing the issue, in conjunction with police, football clubs and stewards, and says all options should be on the table for punishing the individuals responsible and also for clubs failing to ensure player safety.
"These are criminal acts," Taylor said. "If it happened on the high street there would be severe action taken. There is no reason why that should not be done just because it is a football ground.
"There is a combination of sanctions [for clubs] with fines, ground closures, games behind closed doors and points deductions. Every case will be looked at on its merits.
"It has almost been a copycat of what has been happening. We have had three incidents this weekend. We have had other incidents this season.
"It seems to be a time when these things are coming back. We need to improve the atmosphere.
"You can have commitment and enjoyment and it is great entertainment but when it bubbles over to this it is no longer what we want it to be. It is about setting an example.
"We have been warned now on too many occasions. This has got to be dealt with by the football bodies, the police, the stewards and the clubs. There is a total responsibility.
"We have got to address it now. If not, it is going to be too late and there is going to be a tragedy."
Sports Minister Davies added in a statement: "The incidents that happened over the weekend were a disgrace. It is right that they are investigated immediately and strong action is taken by the football authorities and clubs.
"Protecting players' safety is of paramount importance and every possible step must be taken to ensure that they are not put at risk."