Everton are "the worst run club" in England and "demonised" their supporters after alleging safety concerns for board members at Goodison Park, according to Jamie Carragher.

The Toffees' board of directors did not attend the January 14 home loss to Southampton upon the advice of security professionals, citing a "real and credible threat to their safety and security".

Everton released a statement before kick-off to explain the situation was caused by "malicious and unacceptable threatening correspondence received by the club".

The Toffees' struggles culminated in Monday's sacking of manager Frank Lampard, but Carragher – an Everton fan growing up – focused his ire on the club's handling of the supporters' discontent.

"There was a protest that everyone knew about, it wasn't going to be during the game it was going to be at the end of the game," former Liverpool captain Carragher told Sky Sports. 

"Everton's board put out a statement saying there had been threats, and I'm not denying that – and if it is true then that's bang out of order.

"But I think someone should have been at the game, the statement they put out as a football club was bang out of order. To me, they demonised the whole fanbase, and I'm not saying they are telling lies in terms of threats, but that is the name of the game."

When concerns over physical threats and violence were put to him, Carragher responded: "There may have been but there was nothing reported to the police, you need to say that too.

"They threw the whole fanbase under the bus, you don't do that, especially in the city we are from. In Liverpool, you back your own whether it's your family or your football club. That's what you do.

"[The supporters] were there to criticise them and get them out of the club, but those supporters kept them up last year, don't forget that.

"Supporters are protesting that they want them out of the club, [the board] felt they couldn't go to that game. On the back of putting that statement out, how do they go back to the game or to Goodison. Lampard is gone but their banners aren't going anywhere until those people have left the club. 

"Until there is serious change there, they're not going to get the fans back on side and it was them who kept them up."

Everton sit 19th in the Premier League on 15 points following a 2-0 defeat to fellow strugglers West Ham on Saturday, their eighth loss in their last nine matches in all competitions.

The Toffees' next manager will be their seventh permanent appointment since Farhad Moshiri – who was in attendance at London Stadium – invested in the club in 2016.

Carragher believes dismissing Lampard was the right decision, though his concerns remain over Moshiri's relationship with board members Bill Kenwright and Denise Barrett-Baxendale.

"It was right to change the manager, but no one knows a football club better than their own supporters," he added. "Their own supporters haven't got banners for Lampard, they've got banners for Moshiri and the board.

"I said this about six months ago, Everton are the worst run club in the country. I'm not saying that as an ex-Liverpool player, I'm saying that as an ex-Everton fan.

"When I made that comment, Everton actually got in touch with me and I thought fair enough, you're saying things in the media and sometimes things come back to you. But I didn't think I was wrong when I said it then and I don't think I'm wrong now.

"Moshiri doesn't know what he's doing but he's got a lot of money – and he's put a lot of money in. Why does every Everton manager fail? You've got to look at the top, and that's Moshiri. It's a mess.

"There's a massive divide between Moshiri, Kenwright and Denise Baxendale – what is the role of Kenwright? It's not financial and the owner is not listening to him. If they are there for expertise, they're not listening to them, why have them there?"