The Football Association is confident of persuading Gareth Southgate to extend his contract as England manager beyond 2020 after he unexpectedly led the team to a World Cup semi-final in Russia.

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Martin Glenn, the FA chief executive, is aware the governing body cannot compete with the financial lure of Premier League management but believes Southgate shares his view that there is “unfinished business” with the national team. Southgate’s deal ends in 2020 and, providing England qualify for the European Championship that summer, it is understood the FA would be keen to extend it until at least the 2022 World Cup in Qatar.

“We’d like him to stay beyond 2020 and I think he would like that too but we haven’t spoken about it in any depth because that would be a contract discussion and he’s on holiday now so we’ll talk about that when he gets back,” Glenn said. “It’s our job to make it competitive to make sure him and all of our people are competitively rewarded. We can never compete with a Premier League club in terms of pay.”

England reached a World Cup semi-final for the first time since 1990 in Russia before being defeated by Croatia in extra time. They also won a World Cup penalty shootout for the first time, beating Colombia in the last 16. Much of that success was attributed to the fastidious preparation carried out by Southgate and his lieutenants and an improved team spirit.

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“He’s really blossomed,” Glenn said. “It was fantastic seeing him first hand in pressure situations. He fits the definition of the modern manager we want. He gets that it’s not just picking the team but all the aspects behind it and has benefited from a world-class support team. I’m sure he’d be the first to say that.”

Glenn claimed the FA is entering a new chapter following triumphs at the under-20s and under-17s World Cups and the possible sale of Wembley which could lead to a huge injection of cash into grassroots facilities.

“The World Cup makes our work feel justified. It’s reinforced a belief we’re doing a lot right,” Glenn said. “Our target is to be a top-four team in the world and we’re not there yet, so there’s unfinished business.”