He looks set to be out of action for the months of March and April but is hoping to be back for Connacht's concluding games of the season at home to Cardiff and the Ospreys.
Swift explained: "I was told by the doctors that I had two tears on my kidney and that the earliest I'd be back playing is eight weeks time, so there's a possibility I might not play again this season.
"I'm very disappointed as the season was going very well for me and I'd played in every game since we played South Africa in August.
"The important thing now is to focus on recovering and getting back to training but for now I've been told to get plenty of rest."
The London-born forward, who started off life in the back row, has played in all of Connacht's 18 competitive games this season and has made a total of 131 appearances for the westerners to date.