Terry has a number of options to consider following his decision to leave Chelsea after 22 memorable years with the west London club.
The 36-year-old has a strong relationship with Swansea boss Paul Clement following their time together at Stamford Bridge and that could be a major advantage when it comes to his final decision.
Swansea chairman Huw Jenkins says the club are in the market to sign quality players this summer and remains confident they can win the race to clinch Terry’s signature.
Jenkins said: “I know Paul has spoken to John and his agent. I don’t know what his preferences are because he has options and will go with what suits his family.
“We are well covered in that area and quite strong but I know there is interest with Paul and he spoke with John and his agent himself.
“When Paul knows a players well it is an obvious one for us. Sometimes it works and sometimes it doesn’t but we will always try and do that.”
Jenkins is set to be a busy man this summer as speculation continues to surround the future of star midfielder Glyfi Sigurdsson.
Everton have been strongly linked with a move for the Iceland international and the Swansea chief insists it will be a big challenge for them to keep him at the Liberty Stadium.
Jenkins added: “It is a challenge for clubs like us. It is a challenge for every club to keep their best players. We have never gone actively out to sell players and we are not going to do that now.
“It is natural when Glyfi has had a good season with goals and assists and there is always media interest.
“There will be one or two teams looking at our players and we have to accept that. That is the Premier League and we have to deal with that.
“Paul has a big season ahead of him and the ideal situation is we still have Glyfi here for the first game. I have had no offers for him so far and no serious discussions.
“You get agents ringing around but we are not actively looking for anything. We know the reality of the market and will have to see what happens.”
Jenkins has confirmed the club are in the market to sign Chelsea forward Tammy Abraham, but face stiff competition from newly promoted Newcastle and Brighton.
“Tammy is another player who Paul knows well and I am sure the player is weighing up his options.
“Chelsea have agreed for him to go out on loan and I am led to believe that both Brighton and Newcastle are interested alongside us.
“We would be keen to do something and he will weigh up his options when he returns from England Under-21 duty. We are eagerly awaiting what he wants to do next.”
Jenkins was Speaking at Swansea City’s launch of their new Letou shirt sponsorship.
As part of their sponsorship, the new sponsors have launched “Letou Goals for Good” and donated their entire corporate hospitality allocation to Disability Sport Wales and Shelter Cymru in order to give something back to the community.