Southgate is more optimistic about the fitness of Dele Alli, who returned to the starting line-up against Colombia after missing the Group G games against Panama and Belgium with a thigh problem.
England beat Colombia 4-3 on penalty kicks and will face Sweden in the quarterfinals in Samara on Saturday.
Alongside an Instagram picture of him sat on a jet, Delph wrote: "I've just experienced the most incredible 24 hours".
Southgate has challenged his players to write their own history after England's poor recent results at finals.
"At 07:52, Me, my wife and two attractive daughters welcomed there [sic] baby sister into the world". "We went to the Euros (in France) two years ago but saw the violence between Russian fans and England fans, so that put us off going to the World Cup". "Back to Russian Federation now".
"I think it's an important moment for everybody in our country really", Southgate said after the penalty win.
"I just went through the process that I normally do, that I've been working on", he said of his effort on Tuesday night, which was saved to the goalkeeper's left.
In a tweet signed off by William posted on the official Kensington Palace page, he said: "I couldn't be prouder of #England- a victory in a penalty shootout!" That's the key in football. Every team will believe they can make the final at this point.
"It's been a fantastic year for me personally, to achieve everything I have achieved", he said.
A bare-chested, grinning Dier was labelled the "Coolest Man in Moscow" by the Daily Mail, who added with a touch of hyperbole: "The explosion of relief must have been heard in space".
"That game's gone now, I don't really want to comment".
"I grew up in Portugal, so I'm very aware of it [England's poor penalty record]".