Ms Jayalalitha, a powerful and controversial politician, has been in hospital since 22 September.
Anxiety about her health has grown in the absence of clear information about her condition.
There are concerns that her supporters could resort to violence and acts of self harm in the event of bad news.
There is a veil of secrecy over the extent of Ms Jayalalitha's illness which has triggered a wave of rumours, observers say.
The two arrests over the rumours were made in the state capital Chennai (Madras) on Tuesday. Cases have been registered against 40 others.
Why ailing leader has Indian state on edge
The downfall of India's 'Mother' politician
Profile: Jayaram Jayalalitha
Police said that one of those arrested was a technician who had uploaded an audio clip to a website he managed, of what he claimed was a hospital official talking about Ms Jayalalitha. The other man, an engineer, was arrested for posting "false information" on Facebook about the chief minister's health,  The Times of India said.
The hospital treating the chief minister initially said she had "fever and dehydration", but in a media release on 3 October said that she was on "respiratory support" and was being administered antibiotics. A British doctor specialising in critical care treatment has also been flown in to examine her.
A photograph of a woman in an intensive care unit went viral on social media, with many claiming it was of Ms Jayalalitha. It was later found to be a picture of a hospital in Peru.
Indian police have arrested two people for "spreading rumours" about the health of Jayaram Jayalalitha, chief minister of southern Tamil Nadu state.