Altidore, who spent last season on loan at Hull from Villarreal, scored six goals in the qualification campaign but there were concerns over his fitness after an ankle problem forced him out of a friendly win against Australia at the weekend.
However, Bradley insists there was never any real doubt about the 20-year-old and, as he has returned to full training, he could well be in the starting line-up in Rustenburg.
"The injury cost him a couple of days (of training) but, from the start, we knew it was very minor," he said. "The fact he is back in regular training means he is available for whatever role we choose on Saturday."
Defender Onyewu, meanwhile, saw his first season at AC Milan ruined by a patellar tendon injury seven months ago, but although he has not made an appearance in any of his country's warm-up games, Bradley feels he is ready to return.
"The fitness programme that we have put our players through for the World Cup means all 23 players are ready to play 90 minutes on Saturday if asked," he said. "It is a credit to our staff and the players because they have worked incredibly hard."
Bradley also revealed that Sir Alex Ferguson has given him guidance as he prepares for his first World Cup.
The former Chicago Fire, MetroStars and Chivas USA boss, who took charge of the American national team in 2006, has paid visits to a number of Premier League clubs and says he has learned a lot about top-level management from some of the greatest coaches in the game.
"I have enjoyed the opportunity to get to certain clubs, to see how things work," he said. "Without a doubt there have been some managers who, when you get to know them a bit, have been great in being open.
"You find they know a great deal about soccer in the United States, they know about our national team. You get a sense that the environment inside the club often has a lot to do with the personality of the manager.
"The times I've been at Manchester United, I appreciate the way Sir Alex has set a whole tone within the club. There is a down-to-earth way when sitting in the cafeteria and you see the players that have been there.
"Whether it is Giggs, Scholes, or the young reserve players, you see the way things are done. It is a big club but at the same time there is a down-to-earth quality, a realness, a part to it that for me, I think is important."
Bradley says he also had the chance to learn from his first World Cup opponent, Fabio Capello, during the Italian's time at AC Milan in the 1990s.
"I also watched AC Milan a great deal in their great years which Fabio Capello was part of. That even included taking a college team to Milan at one point.
"I have certainly seen his work over the years, whether it was Milan, Real Madrid, Juventus, England. His career speaks for itself. For me, I learn something every chance I have to see how these people work.
"Whenever you come up against teams that are managed by top managers like Capello, you expect they will be well prepared, tactically and mentally right.
"He sets a good tone with his team about how they need to play, how they go about their business. For me the most impressive thing when you look at Capello's career is how he has adjusted to the different teams, to the talents, tweaked things.
"When he took over at Milan, he kept a lot of the good things but tweaked it a bit with the number of games Milan were playing so the physical side could be managed.
"In later years at Roma, it became more of a 3-4-1-2 and the way the system worked with a player like Cafu on one side.
"Capello's quote to me on many occasions was 'when you make wine, the grapes aren't always the same'. At first, I thought he was on about wine but I realised he was trying to tell me something about football!"