The former Tottenham Hotspur boss joined the club in November last year and was given full control of football matters while working alongside former head coach Luke Williams.
His arrival was not enough to help retain Swindon's League One status as they suffered relegation after picking up 44 points from 46 games.
"No, Tim's gone. That [director of football role] has gone," the Daily Mail quotes Power as saying. "I think we needed a change, it's a different set of circumstances.
"When I came into the club it was losing £5m a year and we need to find a way of plugging that hole. I went into recruitment and found players we then sold to plug that hole.
"That's getting harder and harder so, especially in this league, we need a little more experience than what we've had in the past."
David Flitcroft replaced Williams as head coach earlier this month.