Southgate was appointed England boss in 2016 and has guided the team to World Cup semi-finals and quarter-finals as well as runners-up at Euro 2020.
His contract is due to expire at the end of the year, with Manchester United said to be interested if Southgate chooses to leave the national team post after his fourth major tournament.
The 53-year-old has previously said he has shelved any discussion of a contract extension until after this summer’s tournament, and Pickford suggested the lack of certainty has not affected the squad.
‘I think there’s no difference. I think there’s one goal and that’s to win a European trophy,’ the Everton goalkeeper told BBC Radio 5 Live.
‘Us as players, and I think the manager and the staff, that’s all our goal is, to be focused on winning the European Championship. I think all the noise around anything else will probably come after that.
‘His character has not changed. He’s a top man manager and he’s got a great set of staff around him. The training’s always good with the manager and everything we do is good.’
Southgate is closing in on surpassing Bobby Robson to becoming the third-longest-serving England manager in history.
Despite his lengthy tenure, though, many prominent figures within the England camp would be happy to see him continue.
‘If Gareth’s the man to make us be successful then we want him to stay as long as possible,’ England centre-back Harry Maguire said on Thursday.
‘I’ve had a great relationship with him. I’ve been a big part of the success and the progression that we’ve shown as a country.
‘And I’m sure he will say the same thing now, it’s all about winning a trophy. I don’t know what’s going to happen after the Euros, I don’t know if he knows what’s going to happen.
‘But I feel like what he’s done from when he first took over this job to where we are at now, we’ve taken huge strides and he’s a massive part of that.
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