Southgate encouraged by progress of injured Shaw and Maguire before Euros
Maguire and Shaw missed Manchester United's FA Cup final win against Manchester City last weekend as they recover from muscle problems.
Southgate included both players in his provisional squad for the Euros, but the England boss needed to see signs of improvement in their health before next week's deadline to finalise his 26-man group for the tournament in Germany.
Maguire, a key figure at centre-back given England's lack of options in that position, has been out for a month, while left-back Shaw has spent three months on the sidelines.
The pair will miss Monday's friendly at St James' Park against Bosnia and Herzegovina and are unlikely to feature in Iceland's visit to Wembley on Friday.
But Southgate said Maguire and Shaw, along with Newcastle forward Anthony Gordon, who was also absent from training, are all showing good signs in their recoveries.
"We're pleased with the progress they've made. None of them will be involved tomorrow," Southgate told reporters on Sunday.
"I think Anthony has a good chance to be available for Friday. The other two a bit more unlikely, but we will just see how quickly they can go really.
"Same tomorrow for Bukayo (Saka), probably the same with John Stones. Everybody else will be involved.
"John because he reported a bit later, so he hasn't worked with us for the number of days so better for him to follow an individual programme for the next couple of days.
"Bukayo is fine, he's progressing well but the periodisation we've taken from his injury is to rest tomorrow and then he should be available for Friday."
'An incredible year'
Manchester United's FA Cup final goalscorer Kobbie Mainoo and Manchester City duo Kyle Walker and Phil Foden are yet to link up with the squad.
Real Madrid star Jude Bellingham will be another late arrival to the pre-tournament camp following his club's Champions League triumph.
The 20-year-old's dream first season at Real climaxed on Saturday as the Spanish club beat his former team Borussia Dortmund 2-0 in the Champions League final at Wembley.
Bellingham, who set up Vinicius Junior to score Real's second goal in the final, will be vital to England's bid to win their first major trophy since the 1966 World Cup.
And Southgate saluted the midfielder's superb form, which saw him named the La Liga Player of the Season after scoring 23 goals in all competitions.
"He's had a fantastic season. To join the biggest club in the world and perform the way he has throughout the season has been an incredible year for him," he said.
"Absolutely delighted for him to win last night. You saw him with his family and they have to take enormous credit, he is as he comes across in his interviews.
"He's authentic, that's how he is every day, whoever he is interacting with. He's got the humility and understanding.
"For him to cap that season with winning a Champions League has been a phenomenal year for him."
England, one of the leading contenders to win the Euros, kick off their campaign against Serbia on June 16 before facing Denmark and Slovenia in their other Group C fixtures.
