It is understood that Celtic handed over a sum of around £1.3m for the 29-year-old, using young midfielder Christie as a makeweight to push through a transfer.
After finalising terms to remain in Glasgow until 2020, Hayes admitted that he cannot wait to get going next season.
"This is a huge honour for me. It is a dream come true to be part of this brilliant football club and I can't wait to get started," he told his new club's official website.
"I'm delighted. It's been in the pipeline for a little while now but I'm pleased to finally get here. I was eager to get it over the line for the last few weeks.
"Celtic have done well over the last few seasons and with the manager coming in last year he raised the level again. It was hard last year seeing all the celebrations and all the stuff on social media and you want that for yourself and to be part of it.
"Now it's my chance and hopefully, going forward, I can create my own little bit of history and take the chance I've been given."
Hayes was the only non-Celtic player to be nominated by his peers in the SPFA Player of the Year awards for 2016-17.