The FFR made the decision due to ongoing uncertainty over the availability of the Stade de France, although it had yet to officially the name the city or venue that would stage the event. European Rugby Cup (ERC) subsequently announced it was looking for other cities to tender for the event with a decision to be made by the end of the month.
The WRU has put itself forward and proposes to stage the Heineken Cup final at the Millennium Stadium with the Cardiff City Stadium hosting the Amlin Challenge Cup decider.
A formal approach has been made by WRU chief executive Roger Lewis to confirm the offer which, if successful, would see Wales host the Heineken Cup final for the seventh time, four of which have been at the Millennium Stadium.
Lewis said: "We are proud of our Heineken Cup heritage in Wales and would be delighted to host the 2014 final at the Millennium Stadium.
"We know supporters, players and coaches savour the rugby experience Cardiff and the Millennium Stadium can offer and we would work tirelessly to make the game yet another great success.
"We will now be liaising closely with the Welsh Government and Cardiff council to ensure we can deliver the best possible event for this great tournament if our bid is successful.
"I am desperately sorry for the FFR as they would undoubtedly have provided a great stage for the final but I applaud the decisive action they have taken to ensure we have time to put a proper plan in place for 2014.
"The Heineken Cup is a jewel in the crown of the club game and the finals are always fantastic sporting occasions.
"Rugby is the national sport of Wales and all the previous finals hosted in Cardiff have been superb occasions."