The 35-year-old has represented the club more than 500 times over the course of two spells in south Wales and his playing career has slowed over the last few seasons.
Britton’s game time has been limited by injuries and he previously suggested that he had an interest in moving into coaching.
The midfielder will now get his first taste after being handed a dual role as both player and assistant coach, while a new club captain will be named.
Clement had the vacancy in his backroom staff following Makelele’s decision to leave that post and take up the managerial position with Belgian top-flight outfit K.A.S Eupen.
Swansea confirmed the news in a statement: "While the 35-year-old remains an important part of the playing squad, he will also work under manager Paul Clement and alongside Nigel Gibbs, Karl Halabi and Tony Roberts," it read.
"Britton will train and be available for selection as normal, however, he will be part of the coaching staff when he is unavailable to train or play."