The Ghana international has made 63 appearances since joining the Danish side in 2014 and will join the Premier League high-fliers pending an agreement of personal terms.
FC Copenhagen chief executive Anders Horsholt did not specify a figure but admitted the offer was too good to refuse.
"We cannot turn down a transfer of this size because, generally speaking, it is the best outcome for Copenhagen," Horsholt said in a statement.
"We have been very happy to have Daniel here but it is part of our strategy, as a club and a company, to develop players to a level where they get so attractive that clubs from the biggest leagues will buy them for large sums of money."
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The former Wolves boss added that Leicester have been long-term admirers of the 21-year-old.
"From the first day Daniel has been a role model and a key player for us, and he has improved dramatically and faster than anyone could have expected," he said.
"He has handled his success and all the attention extremely well, and it is a dream come true for him to play in the Premier League for a club who has targeted him for a long period.
"We wish Daniel all the best in the Premier League and we are sure he will have a great career."