The Devils are entitled to compensation should any of the players sign elsewhere.
Forwards Beau Bennett, Devante Smith-Pelly and defenseman Jon Merrill did not receive offers, although the team is hoping to re-sign them before they become unrestricted free agents on Friday or after that.
New Jersey did not made a qualifying offer to defenseman Raman Hrabarenka, who played last season at Albany along with Scarlett.
Palmieri had 30 goals and 27 assists in 82 games in his first season with the Devils. Boucher had eight goals and 11 assists in 39 games, while Josefson had four goals and 10 assists in 58 games.
Smith-Pelly had eight goals and five assists after being acquired from Montreal. Merrill was limited to 47 games by injury.
Bennett was acquired Saturday from Pittsburgh for a third-round pick. He had six goals and six assists in 33 games for the Stanley Cup champions.
Also Monday, the Devils named Dan MacKinnon their senior director of player personnel. He will oversee all aspects of the club's pro scouting operations. He also will work with the organization's hockey operations and pro/amateur scouting staffs regarding player evaluation, acquisition and draft preparation.