The Manchester United legend had an ill-fated 31-day spell in charge of fellow fourth-tier club Oldham Athletic last year.
He is stepping in on a caretaker basis after Salford, unbeaten in the league this season, decided to terminate Alexander's contract after a reign that started in May 2018 and included promotion from the Conference National.
Scholes' first game at the helm of Salford, who surrendered a two-goal lead to draw 2-2 with Tranmere Rovers on Saturday, will come at Port Vale this weekend.
The 45-year-old is one of the part-owners of Salford, with former United team-mates David Beckham, Ryan Giggs, Nicky Butt and Gary and Phil Neville also invested in the club.
Eriksson became the first non-British manager to be appointed England boss when he left Lazio to take over in 2001, turning around their qualification campaign for the 2002 World Cup and securing a spot at the tournament in Japan and South Korea.
With the likes of Rio Ferdinand, Frank Lampard, Steven Gerrard, Paul Scholes, David Beckham and Michael Owen, the Three Lions were fancied to make an impact.
But they were beaten by eventual champions Brazil in the last eight. Two years later they fell at the same hurdle in Euro 2004 to Portugal, who also sent England packing in Eriksson's final tournament in charge – the 2006 World Cup.
One of the main legacies of Eriksson's time in charge was a perceived inability to get the best out of England's so-called 'golden generation', but he thinks Southgate has more options at his disposal.
When asked if he feels the current England team was better than his, Eriksson told Stats Perform News: "Maybe not, but they have more choices today than we had.
"Number 20, number 21, the quality went down a bit. It's easier today, there are many hugely talented football players.
"They did well at the last World Cup, they will be even better in the next Euros. It's a new generation. They are young, they are good.
"They have a lot of quality all over. It looks like a very hungry team. They have a lot of pace and that's important.
"If you defend well, then you will be very strong in counterattacks and then, you have a born goalscorer [Harry Kane], and you need that man who can score [many] goals in a major tournament."
Recently, Eriksson was criticised by Ferdinand for apparently urging the former Manchester United centre-back to not play out from the back – but the Swede insists that was not the case.
"He was one of the best central defenders in the world, maybe the best, and as he rightly said, he could play," Eriksson added. "He was a very modern central defender. Football was different then.
"Everybody wants to play like Barcelona, but not everybody should. Not everyone can play like Ferdinand. But that back four was very, very strong.
"I always in all my career was very keen not to lose the ball when we have it in our own half of the pitch: you give opportunities to your opponents to create.
"If you're going to lose it, then do so up front, but if we could play in a secure way from behind, then do it, if not then don't make life difficult for us. But I never ever said don't play from the back."
Euro 2020 was due to start on Friday, but due to the coronavirus pandemic it was postponed for 12 months in March.
England will face Croatia, Czech Republic and as yet undetermined third team, who will be decided by the qualification play-off, when the tournament takes place from June 11, 2021.
Ole Gunnar Solskjaer's men squandered numerous chances in a 2-1 loss to Sevilla in Cologne.
United have been heavily linked with a move for Borussia Dortmund star Sancho, who scored 20 goals in 2019-20.
Former United star Scholes believes the Premier League club need to spend money on an attacker, saying Sancho would have delivered in Germany.
"I don't think they played badly, I thought they were really good against a really good Sevilla team, but that 15, 20 minutes after half-time, with all them chances, you've got to score them," Scholes told BT Sport.
"We're talking about quality, we talk about spending money on centre-forwards and wide players, and you talk about hundreds of millions these days to get these players, but these are the players that win you trophies, these are the ones that win you medals.
"We know all this Sancho thing is going on, but they need to start spending the money. If they want to win trophies, they've got to start spending the money.
"Okay, created chances tonight, but if Sancho was in that team tonight I think United are scoring."
Winners of three Europa League titles since 2014, Sevilla will face either Inter or Shakhtar Donetsk in the final on Friday.
