Swann said the Blues and North had already started working on a Good Friday proposal that they would present to the AFL when they submit their fixture requests later this year.
Swann's comments follow North chief executive Carl Dilena's revelation on Wednesday night that another club had approached North and Carlton separately, offering to play each on Good Friday.
"Between ourselves and North we're really keen to do it. We're certainly going to present a case with North to play a game on Good Friday next year," Swann said.
"We're with North on this one and we've already had meetings about how the game might look.
"We've got a good relationship with North and we've worked well together to make our recent Friday night games bigger events and the crowds have been good.
"We just think this is now the next step."
Swann acknowledged other clubs would almost certainly put up their hands to be part of a Good Friday game, but said Carlton and North would be pushing their case hard.
"I would think for sure there will be other teams keen to be part of Good Friday football if it gets the go-ahead," Swann said.
"If you look at the calendar, it's the last blockbuster space that's available I would have thought.
"So if you can get to play in it and you're the first cab off the rank, then we've seen how it works with the other big blockbuster games, it seems the teams involved stay the same and away you go."
Swann agreed with North chairman James Brayshaw's comments on Wednesday night that Andrew Demetriou's recent resignation as AFL CEO would make the introduction of a Good Friday game more likely.
The Blues CEO acknowledged there would be opponents to a game being played on one of the most sacred days on the Christian calendar, when the longstanding Good Friday Appeal for the Melbourne Children's Hospital is also held.
But Swann said Carlton and North would work closely with the Appeal organisers, including its broadcaster Channel Seven, to ensure the game enhanced rather than detracted from the day.
"From the Appeal perspective, we just feel that we can make a meaningful contribution to it," Swann said.
"We would seek to tie the game in with the Appeal and do whatever's best suited to actually make it work."
