England spokesman Mark Whittle said that Defoe should be back in time to be available for selection for England's opening Group D fixture against France in Donetsk on Monday.
He is widely expected to be selected as an impact substitute.
Defoe published a message of thanks to his well-wishers via his Twitter account on Friday.
He said: "I want to thank the fans, my family, friends, the England boys, the FA and manager for their support.
"I can't tell you how much it means to me. I love you all, may God bless you. A special thank you to Royal Marsden's doctors & nurses who looked after my dad. Very special people."
Defoe won his 47th cap last Saturday when he came on as a substitute and hit the post in England's 1-0 win over Belgium in their final warm-up match for the tournament.
He has won 30 of his England caps as a substitute and has scored 15 goals.
More than 3,500 spectators, including hundreds of children, attended a leisurely England training session in warm sunshine at the Suche Stawy Municipal Stadium, in the outskirts of Krakow, on Friday morning.
In all, according to informed local estimates, more than 5,000 people wanted to be inside -- so at least 1,500 were left out to try and translate a slogan on the side of the bus that declared 'One prize, two nations, three lions.'
On day two of England's new and more open approach to tournaments, there were fanfares, chants, cheers and, most importantly for manager Roy Hodgson, no further injuries.
All 22 available players, minus Defoe, were involved in the training session, albeit that nine chose to disappear into the gym rather than work out in public on the pristine playing surface.
"We are all well, all fit with a clean bill of health," said Whittle. "No injuries. No problems at all."
On a warm day, the players were greeted with a guard of honor as they walked onto the field, local children from the Siemacha Foundation, a charity that supports orphans and underprivileged children in the city, cheering them by name as they arrived.
Striker Wayne Rooney, who is suspended from England's two opening games, prompted notable cheering with his all-action approach to shooting practice. (Reuters)
