The hosts of the next three World Cups have already been confirmed, starting in Brazil in 2014, before going Russia in 2018 and then Qatar in 2022.
Cardiff is set to host the UEFA Super Cup in 2014, only the second time it has not been played in Monaco.
And First Minister Jones made it clear in a recent major events conference that he doesn't want it stop there.
He said: "We have got the UEFA Super Cup coming to Cardiff next year and we want that to be the starting point for bringing more major football events to Wales.
"We are talking with the FAW about bidding to be one of the venues for the 2020 European Championships, we want a Champions League final... and then there is the World Cup in 2026.
"The most likely joint bid would be with England and we know we have three good stadia we can offer.
"By 2026 we can get the Liberty Stadium and Cardiff City Stadium up to 30,000-plus to supplement the Millennium Stadium."
His ambition has been backed by FAW chief executive Ford, who also spoke at the conference.
He said: "It's a great future ambition. We know where the World Cup is going in 2018 and 2022, but the process thereafter has yet to be determined.
"I absolutely support the First Minister in having a good ambition."
Jones also hopes the success of Welsh superstar, and the world's most expensive player, Gareth Bale, will play into Welsh hands not only in hosting a World Cup but qualifying for one as well.
He said: "Football is the world's biggest sport and we know everyone has been talking about Gareth Bale since his move to Real Madrid and giving Wales a lot of global exposure.
"We hope he can inspire the Welsh football team to qualify for a major tournament in the future.
"You only have to look back to the 1990s, at the Jack Charlton era in Irish football, to see how a football team can build the confidence of a nation."