The Hoops are three points clear of Rangers at the top of the cinch Premiership with five matches remaining.
Day joins from the Rugby Football Union to replace Baroness Sue Campbell
Harry Kewell said his Yokohama F-Marinos players can "handle any pressure" after beating Ulsan Hyundai 5-4 on penalties on Wednesday to set up an Asian Champions League final against Al Ain.Al Ain beat Saudi Arabia's Al Hilal 5-4 on aggregate on Tuesday to reach the final.
Day joins from the Rugby Football Union, where she has worked as a board member, chief operations officer and chief financial officer.