The 18-year-old becomes Neil Lennon's second signing since the end of the season after the arrival of French defender Christopher Jullien from Toulouse.
Connell turned heads after breaking into Bolton's first team in January and won a call-up to Mick McCarthy's Republic of Ireland squad in May, only to be forced out through a thigh injury.
The Liverpool-born left-footed player leaves cash-strapped Bolton as administrators attempt to push through a takeover after the club's relegation to League One.
"It's unbelievable to sign for such a massive club, known all around the world," Connell said.
"It's a dream come true, it's what you dream of when you're a little boy."