The highly-rated duo, and King, took part in a squad meeting with manager Fabio Capello on the pitch at England's first training session following Saturday's disappointing 1-1 draw with the United States but swiftly departed when the remaining 19 players began work.
Although England officials played down their absences, they could not erase the fact that Capello has problems with his goalkeeper, his central defence, possibly his outstanding raiding full-back and one of the tournament's most highly-rated strikers, not to mention a lack of midfield craft.
An England spokesman said King was going to "remain under close observation" this week and had not been ruled out of possible selection for Friday's clash with Algeria in Cape Town.
Capello, in a comment on the England website, said King's injury did not look too serious and he could possibly play against Algeria.
"He has just a little problem," explained Capello, who did not speak to reporters on Monday.
Rooney was rested after being kicked on an ankle but Cole's absence was not explained. Reporters were told that he was too tired to train and needed an additional day's rest to recover.
King was drafted into Capello's team after former captain Rio Ferdinand was ruled out of the World Cup with a knee injury suffered in their first training session.
The absence of all three, even if not due to serious injuries, added to Capello's dilemmas following goalkeeper Robert Green's schoolboy howler on Saturday which gifted the USA an equaliser in the 1-1 draw when an innocuous shot squirmed through his hands.
"I didn't make mistakes and I'm not worried for the future," Capello told reporters on Saturday when asked if he regretted picking Green, King, who has a chronic knee injury, or midfielder James Milner, who was withdrawn after only half an hour.
"We played a good game and, for me, the most important thing was to see the spirit of the team and also the physical condition of the players."
"It is up to the manager, he picks the team," he said. "I think he has played in nearly all 12 games, bar a couple, so he is pretty much th
