All the other games in the tournament will take place at venues in England with the final of the competition being played at Twickenham.
Two of the quarter final matches would be played in Cardiff with the others taking place at stadia in England. All the semi-final games would be played at stadia in England.
A decision on the successful bid is expected to be delivered by the International Rugby Board following a special council meeting on July the 28th.
The Group Chief Executive of the WRU, Roger Lewis, said: "I am delighted to be able to confirm that Wales will play a major role in RWC 2015 if this bid is successful.
"We have a proud history of involvement in the Rugby World Cup and Welsh rugby fans will again be able to support one of the greatest competitions in the sporting calendar.
"Some of the Wales pool matches would take place at the Millennium, but there could also be pool stage games involving other major rugby nations such as France or New Zealand.
"The quarter finals are a critical stage of the cup competition and we know the Millennium Stadium will deliver the kind of rugby atmosphere which will make the game a fantastic spectacle.
"Rugby World Cup matches are watched on television screens in some 70 countries around the globe so Wales will again be showcased on the international sporting stage.
"Rugby is our national sport here in Wales so I am particularly delighted that our fans will get to see key quarter-final matches.
"Fans from around the world now fully appreciate how accessible our city centre stadium is so I am certain we would be welcoming supporters from many nations to watch the tournament games here in our capital city."