The east midlands club announced the 21-year-old, who can play in central midfield or at right-back, had signed a contract until 2020.
"It's a great pleasure to join Leicester," Amartey said.
"I know the club have been interested in signing me for some time and that was a big part of my decision to come here."
Leicester have been the outstanding success story of this Premier League campaign
They survived relegation last season, but remain in the Barclays Premier League title hunt, sitting second but level on points with league leaders Arsenal
"All over Europe, people are talking about Leicester and how well the club is doing in the Premier League," Amartey added.
"The team plays exciting, attacking football and I'm looking forward to being part of it."
Copenhagen signed Amartey from Swedish side Djurgarden in the summer of 2014, and has so far won six senior caps for his country.
The deal remains subject to international and Premier League clearance.
Amartey becomes the Foxes' second signing of the January window after they also brought in winger Demarai Gray from Birmingham for £3.75million.
Source : PA
Source: PA