"Taking into account the development of the football club over the last two-and-a-half years and the image we have portrayed as a footballing side, myself and the board of directors felt it was vital to continue along the same lines," Jenkins told the club's official website.
"It was felt that Paulo Sousa is the right man to do this.
"He will provide Swansea City with a fantastic chance to continue developing and growing over the next few years.
"He's young, bright, enthusiastic and a visionary with extensive knowledge and contacts not only throughout Europe, but world football.
"As a club we feel strongly that we have created something completely different in the British game and become renowned as a quality football team with supporters who really appreciate the way we try and play the game.
"We are committed to keeping this theme moving forward under the leadership and guidance of Paulo Sousa who, I'm pleased to say, has verbally accepted the job.''
Sousa will sign a three-year deal and will be unveiled as the new Swans boss at a press conference next week.