West Ham United are prepared to pay for a huge police presence inside the London Stadium to help prevent a repeat of the disorder that marred the 3-0 defeat by Burnley.

The stadium’s security failings were exposed on Saturday by unrest, which included four pitch invasions and David Sullivan, the club’s co-owner, being struck by a coin during protests against the board by hundreds of furious supporters who easily overpowered stewards stationed in the area below the directors’ box.

West Ham, who are facing heavy punishment from the Football Association, are unhappy with the failure of stewards to deal with the disturbances and have insisted the stadium operator, LS185, places police officers inside the ground in the remaining five home matches of the season. Sadiq Khan, the mayor of London, condemned the trouble on Monday and called for an investigation.

The club are not responsible for footing the police bill under the terms of their agreement with the London Legacy Development Corporation, which pays it in full, but the FA will hold West Ham responsible for what took place.

Board members have been shaken by the protests and have indicated a willingness to make a financial contribution to bolster the security arrangements. The LLDC said it was looking to strengthen security and would welcome financial assistance from the club.

The match against Burnley had been designated as a low-category one by the Safety Advisory Group, which is why officers were stationed outside but not inside the ground. Discussions will take place about whether West Ham home games should now be deemed high-category fixtures, a decision that rests with the police. The next match, against Southampton on 31 March, has been designated as high-category. The stadium could be shut if an agreement over tougher security cannot be reached with the police, though that is considered unlikely.

West Ham, three points above the bottom three with eight matches remaining, launched an investigation and called an emergency meeting with the London Stadium’s stakeholders in the aftermath of the game.

The LLDC and LS185 are aware of a video that appears to show a steward sleeping during the match and questions were raised about the slow response by stewards after Burnley took a 66th-minute lead. A fan immediately ran on to the pitch and was hauled down by Mark Noble, the West Ham captain.

The centre-back James Collins clashed with another pitch invader and a female steward was knocked to the ground when fans gathered below the directors’ box to aim abuse at Sullivan and David Gold, the other co-owner.

There is concern within West Ham’s playing squad that the toxic atmosphere will harm their relegation fight, moreover. Sullivan and Gold were escorted from their seats on safety grounds in the 84th minute, with the former’s glasses saving him from potentially serious injury when a coin hit him.

Missiles were also thrown at supporters sitting in boxes next to the directors. Discussions are taking place about how to strengthen the security around those boxes after protesters were able to run amok in the walkway underneath them.

The buildup to the game had been dominated by the fallout from a cancelled protest march against the board, which featured threats being aimed at Mark Walker, the chairman of the West Ham United Independent Supporters Association, by rival fan groups.

A febrile atmosphere rapidly developed once West Ham fell behind. The LLDC and LS185 are looking at whether there was intelligence to suggest there would be some form of protest during the game. The process will be repeated before the visit of Southampton.

A spokesman for the mayor’s office said: “The disturbances at the London Stadium were disgraceful and it’s clear there cannot be a repeat of the ugly scenes witnessed. It is up to West Ham to carry out a thorough investigation, together with stakeholders, and take proper action against those supporters who misbehaved.”

The FA is studying a report from the referee Lee Mason and is seeking West Ham’s observations. The club expect to receive a fine; the FA’s disciplinary committee has the power to make them play behind closed doors. Aston Villa were fined £200,000 after pitch invasions overshadowed their FA Cup victory over West Bromwich Albion at Villa Park three years ago.