The 49-year-old Neely started his career with the Vancouver Canucks, but was best known as an all-star power forward with the Boston Bruins.
The Comox, B.C., native had three 50-goal seasons in his career and totalled 305 goals and 299 assists in 726 NHL games.
He was inducted into the Hockey Hall of Fame in 2005 and became the Bruins' president in 2010.
The 42-year-old Hamm helped the United States to two World Cup titles, including the inaugural edition in 1991 in China, and won the women's FIFA world player of the year award in 2001 and 2002 — the first two times that honour was given.
Roma is traded on the Milan stock exchange and the nominations were made at a club shareholder meeting.
Hamm is not a club investor, Roma said.
Roma was bought by a group of four Boston executives three years ago and Richard D'Amore, a member of that group, rejoined the board. Former club president Thomas DiBenedetto and New York lawyer Joe Tacopina, who recently became president of second-division club Bologna, left the board.
American investor Stanley Gold was also nominated.
The 13 board members serve three-year terms.