Celtic claim a fee was agreed with Manchester City for John Guidetti prior to them taking the striker on loan.
Xabi Alonso claims no Premier League offers were put to him before leaving Real Madrid for Bayern Munich.
The 18-year-old, who has scored 16 times this season for the Nomads, has signed a deal until the summer of 2015.
"We've brought Rhys in after having looked at the division to see what the best young talent is there - and we think Rhys is one with huge potential," manager Malky Mackay told the club's official website.
"He'll be training with our development squad and under-18s to start with. We'll try and bleed him in gently towards the senior set-up and see how he develops, but he's certainly one with great potential for the future."
Healey added: "I'm really happy to make the move to Cardiff City.
"I'm a striker who likes to play through the middle on the heels of the last defender and I love to score goals."