The 64-year-old minister has a history of chronic diabetes and was admitted to a Delhi hospital earlier this month.
Ms Swaraj is one the most high-profile ministers in Prime Minister Narendra Modi's cabinet.
On Tuesday she tweeted that she was undergoing dialysis after kidney failure.
Unlike many other types of organ donation, it is possible to donate a kidney while still alive because humans only need one kidney to survive.
Social media users responded quickly by offering to donate their own kidneys to Ms Swaraj, with some providing their personal phone numbers.