The current coach, Ben Ryan, is stepping down as his contract expires.
Under Ryan, the team has dominated the world Sevens circuit, and last month won the country's first ever gold medal at the Rio Olympic Games.
The rugby union's chief executive, John O'Connor, extended his gratitude to Ryan, who plans to return to Britain and take a break from rugby.
However, he said Ryan was welcome to re-apply should he have a change of heart.
The position is expected to be advertised next week, along with the position of head coach for women's rugby.