Graeme Jones quickly emerged as one of the candidates for the job but it was thought that talks had broken down between the Wigan assistant boss and Swansea.
Jenkins maintains that he has 'got on well' with everyone he has met, and Sky Sports sources understand that Jones is still Swansea's first choice candidate.
A number of other names have been mentioned in connection with the post, with former Arsenal striker Dennis Bergkamp now the favourite.
The likes of Michael Laudrup and Ian Holloway are also believed to be in the frame, and Jenkins says the hunt is progressing well.
"Only certain people will suit our remit," Jenkins told the South Wales Evening Post.
"There are lots and lots of people who come forward at this stage in the process telling you how they want to help you, but what's important is that we think things through and go about things quietly ourselves.
"The fact that we know exactly what we want for our football club helps a lot.
"There are probably hundreds of people out there who think they can do the job for us, but there are only a few good ones.
"We will only speak to a limited few, and we are very quickly getting to the point of seeing what's right for us."
Jenkins admits that he has considered a number of alternatives since Rodgers left for Liverpool, and has refused to rule Jones out of the running.
"What I tried to do quickly after Brendan went was to speak to a few people and get their thoughts so that we could weigh up our options," said the Swansea chairman.
"That's my job and I tried to do that very quickly.
"The link with Graeme Jones is one of those stories that got out there, but that was just about me quickly weighing up where we were.
"The stories did not come from me or from Graeme, and I won't scratch anyone off the list.
"As always, I have got on well with people in the discussions I have had."
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