The offer to stage the fixture in Cardiff has been made by the WRU following the withdrawal of the bid by the Federation Francaise de Rugby (FFR) because of uncertainty over the availability of the Stade de France stadium.
A formal approach to ERC has been made by the WRU Group Chief Executive, Roger Lewis, to confirm the offer.
If the showpiece game is played in Wales it would be the seventh Heineken Cup final to be staged in Cardiff.
The first two finals were played in Wales before the opening of the Millennium Stadium but since then four more have been hosted at the new home of Welsh rugby.
As the competition has grown staging a final can now provide more than £20m to the wider economy of the host city and its surrounding area.
Roger 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.
“Everyone knows that the Millennium Stadium’s city centre location helps create a great rugby atmosphere which ensures hotels, restaurants and the railway station are just a stroll away.
“The retractable roof also provides any visiting fan with the certainty that the game will go ahead in great playing conditions no matter what the weather.”
The bid would also include staging the final of the second-tier Amlin Cup at the Cardiff City Stadium.
