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Southgate led England to their first major international final since the Three Lions' iconic 1966 World Cup triumph.
However, the England manager's dreams of lifting silverware were crushed after Italy beat the Three Lions on penalties in the Euro 2020 final at Wembley.
Southgate has been praised for his impact as England manager, having previously led the Three Lions to the semi-finals at the 2018 World Cup in Russia.
Despite their defeat to Italy on Sunday, England's youthful squad and progress since 2018 have left fans and pundits optimistic about their chances at the 2022 World Cup in Qatar.
However, Keys has insisted that he can't see England "going any further" under Southgate's leadership.
Speaking in his official blog, the 64-year-old sports presenter ruthlessly slammed Southgate's tactics for the penalty shootout against Italy.
"The whole tournament smacked of Southgate being just a bit too clever," Keys wrote.
"Even the penalties were overthought. I felt for [Marcus] Rashford and [Jadon] Sancho -- deemed not good enough to play a part in 120 minutes -- but asked to go and make fools of themselves in a shootout. Sorry. That was poor.
"So, they were the best in training? So what? Your first touch of the ball on the night is a pen? But I can't see England going any further with Southgate.
"If it was my choice I'd move heaven and earth to persuade Arsene Wenger to take it. He really could end all the hurt."
Wenger, 71, has been out of management since he called it time on his remarkable 22-year tenure at Arsenal in 2018.
However, the legendary Frenchman previously admitted that he turned down the position of England manager more than once because it should be an Englishman in charge.
Wenger told beIN SPORTS in 2018: "Yes, I turned them down two or three times. But I saw it that the daily involvement was, for me, important.
"I felt and said many times it should be an English guy who takes the English national team.
"It's a big country, they have enough quality managers, and I thought maybe it was not right for me to do that."
Marcus Rashford missed his penalty in the shootout and was subjected to vile racist abuse on social media alongside Jadon Sancho and Bukayo Saka.
A mural of the 23-year-old England star in Manchester was also shockingly vandalised not long after his penalty miss in the Euro 2020 final.
However, Rashford has since broken his silence in a powerful statement on social media, saying that he would "never apologise for who I am and where I came from."
Featured Image Credit: PA
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