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The Three Lions suffered an agonising penalty shootout defeat to the Azzurri in the final of the last Euros on home soil in 2021.
Some questioned Southgate's future as England boss after a quarter-final exit to France at the 2022 World Cup.
Ex-Crystal Palace owner Jordan has now claimed he doesn't expect new ideas from Southgate in the upcoming Euros campaign - which could spell the end.
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"I don't believe we have a winner in the dugout," Jordan told talkSPORT.
"So for me, that takes the component that nobody can never program in life, which is that X Factor, which makes somebody different.
"Which makes [Roberto] Mancini be able to out think [Gareth] Southgate on the side of the touchline to win a European championship final.
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"We are all desperate to do things, but it does not mean we can. The key component is we can all be busy falls, but the objective is to be winners.
"And there is only a few of them about, and when you see them, you have to hang on to them for grim death.
"I think it is more of the same (on Southgate continuing where he left off) unless somebody is going to have a personality transplant we are going to see probably the same.
"We can not have the Gareth Southgate apologists and sympathisers, the Danny Mills of the world, that pop out of the ether with their uniformed nonsense, that tell us, ultimately, that he is a winner.
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"With that in mind, winning means winning, we are in moment where we need to win this tournament coming up.
"Anything other than winning it, has to be, with due respect, the end of the conversation.
"England won't win the tournament while I have a hole in my backside, no."