Brighton did not disclose financial details, but British media reported that the deal is worth around £16 million.
Osman, who has four goals and five assists in 29 appearances across all competitions for Nordsjaelland this season, will join the south coast club when the transfer window opens later this year.
"He's a young and exciting prospect, but for now it's important that we give him the space to focus fully on Nordsjaelland, where he is enjoying a good season," Technical Director David Weir said in a statement.
"Once the current season is over and he's had the opportunity to rest during the summer, we will bring him into our environment where he will get the time and support he needs to settle here in England."
Roberto De Zerbi's Brighton, eighth in the top-flight after 23 matches, next visit fifth-placed Tottenham Hotspur later on Saturday.
