OL Group, which owns Lyon's men's and women's teams, said in a statement that the move will consolidate their position as the "top women's football group in the world".
Lyon president Jean-Michel Aulas unveiled at the club's most recent AGM plans to expand his team's brand in both the United States and China and earmarked a budget for the purchase of an American franchise.
Based in Tacoma near Seattle, Reign FC will start their eighth season in the American National Women's Soccer League next term and have on their roster FIFA's women's World Player of the Year Rapinoe.
Rapinoe, 34, was joint top scorer with six goals at this year's women's World Cup - won by the United States - and bagged the Golden Boot.