Highlights - England 1-1 USA
England's Ledley King is expected to be out for three weeks because of the groin strain he suffered against the United States, BBC Sport understands.
That would rule the Tottenham defender out of England's remaining group matches and a potential last-16 clash.
However, England still hope he will play some part in the tournament and he will not be flown home early.
The 29-year-old had to be substituted at half-time in England's 1-1 draw with the US in Rustenburg on Saturday.
King underwent a scan on Sunday that revealed his injury was a "groin muscle strain".
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The Football Association then forwarded the scan on to Spurs for a second opinion.
The centre-back joined his team-mates for the start of England's training session at the Royal Bafokeng Sports Campus on Monday before departing to continue his rehabilitation work in the swimming pool.
England's coaching staff will be relieved that King's injury did not turn out to be more serious, said BBC Radio 5 live football reporter Ian Dennis.
"As a general rule, if it is a grade one groin tear it would mean he is out for three weeks. If it's a grade two then it is four to six weeks and if it's a complete rupture it would mean surgery," he stated.
Dennis also revealed that King suffered his injury "in the third or fourth minute of the match so he could easily have aggravated it in that period".
King's injury leaves England boss Fabio Capello with selection issues at centre-half after Manchester United's Rio Ferdinand was ruled out of the tournament before it began.
However, Liverpool's Jamie Carragher, who replaced King against the US, believes Capello has adequate cover to call on.
West Ham's Matthew Upson and Tottenham's Michael Dawson are the other recognised centre-backs who could partner former captain John Terry for England's second Group C game against Algeria on Friday.
Carragher said: "Losing Ledley is a blow. He is a top player and it is unfortunate because it is the same position as Rio.
"You can't bring anyone else out [to South Africa] now, so any injury is going to hurt the squad but the manager picked five centre-backs in the squad because he knew there could be problems and his selections have been proved right."
Meanwhile, midfielder Gareth Barry insists he will be "spot on" for England's World Cup clash with Algeria on Friday.
The Manchester City player revealed his ankle, which he injured against Tottenham in early May, is fully recovered.
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"The ankle has been fine for 10 days," he told BBC Sport. "Another week's training in the lead-up to the Algeria game and I should be spot on.
"If the manager had asked me if I was ready to play against the United States at the weekend, I would have bitten his hand off to try and make the starting team.
"I was desperate for us to go two or three goals up and then he might have thrown me on."
Barry admitted it had been "touch and go" as to whether he would make the squad at one point.
"It was a really tough few weeks sitting at home every day, and I couldn't do much," he said. "Eventually I was happy to make it [the squad]."
After this week's game against Algeria, England face Slovenia in their final group match on 23 June.
If England make it through the group stages, their last-16 clash would be on either the 26 or 27 June - depending on whether Capello's men were winners or runners-up in the table.
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England cannot now call upon another player to join their squad because any replacements could only be made up to 24 hours before a nation's first game.
King has had long-standing problems with his knee, limiting his ability to train as often as other members of the squad and play back-to-back matches.
But those fears were allayed after the Londoner came through three games in a week for Spurs towards the conclusion of the 2009/2010 Premier League season.
Striker Wayne Rooney did not take part in Monday's session because of an ankle knock.
The Manchester United player and left-back Ashley Cole appeared for training with the rest of the squad but after an initial address by Capello, they and King headed indoors to work in the swimming pool.
Rooney's injury is not thought to be serious, while Cole has been missing full training after a game to help his recovery from a broken ankle he suffered in February.