The former politician and his wife Yvette Cooper are housing a mum and her two daughters after they fled the Ukraine amid the conflict with Russia, and the family's friends are apparently staying nearby.
"They have been living with us in Castleford since last April," Balls said on the ITV show.
"They had to escape overnight when Kyiv was being bombed - they had to leave their dad."
He told The Sun: "Their dad and grandparents are still in Ukraine so they have been back to visit but the continued shelling and power shortages make things very hard."
Two of the girls - who are named Yeva and Zakhar - are champion dancers and have been taking part in competitions.
The paper quoted Yvette as saying: "Yeva and Zakhar are amazing dancers and it’s been inspiring to see them training every day and winning competitions despite everything that has happened."
A source said that Strictly star Jones, who Balls danced with on the BBC ballroom contest in 2016, "has been helping the girls with dance tips".
"It’s been really sweet," they added.
Balls said last year that he was hoping to host a family fleeing Ukraine, saying on Good Morning Britain that he and his wife had applied to do so early on.
After the scheme to house people from Ukraine was launched, several celebrities said they would like to open their homes to refugees, including Sherlock star Benedict Cumberbatch and TV presenter Gary Lineker. Chris Tarrant was one of the stars to open his home to a family.
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