The organisation has spearheaded the scheme to encourage the coffee shops and cafe owners to sell only produce which give farmers a realistic price for their crops.
The charity says Swansea's coffee drinkers can play their part in saving millions of farmers throughout the world from financial ruin.
Claire Gilfedder, Oxfam's manager in Swansea says making the local council join the scheme is a key element.
She said: "We want to lobby all the interested parties and we need the council to introduce it as part of their policy.
"To achieve fair trade status we must get a certain number of cafes and shops to participate and even the council to have fair trade coffee available to all its staff.
"It is a very exciting prospect if Swansea gets there first especially for young people to realise how important it is.
"It will be a big step towards Wales becoming the first fair trade country in the UK."
A spokesman for Swansea council said: "This fair trade campaign is in line with our sustainable policy and we would like to be involved.
"We are in discussion with Oxfam on the whole issue.
"Reports and submissions will have to be made before any firm decision can be taken."
Oxfam claims there is a crisis in the world coffee industry and the launch in Swansea is part of the charity's 'Coffee Rescue Plan'.
Lower prices
Craig Owen, from Oxfam Cymru, said: "Massive overproduction has pushed prices down to an all-time low with 25 million coffee producers facing ruin.
"Four large coffee companies buy nearly half the world's crop and make huge profits but we calculate only around 5% of the retail price reaches the farmer.
"Coffee now costs more to grow and pick than it does to sell."
The fair trade movement says it tries to ensure the farmers are guaranteed a price which provides them with a living income.