The arrests were made following a complaint by an AIADMK woman worker Sunitha Devi, they said.
Sunitha had gone to the bank at Thondamuthur on October 7 when the two staff Suresh and Ramesh sought to know from her about the health condition of Jayalalithaa.
When she did not reply, the two reportedly made some comments on the health status, police said.
A few AIADMK workers along with Sunitha returned to the bank yesterday, as the bank remained closed for five days on account of holidays, and sought an explanation from the two about their comments, leading to some arguments, police said.
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Sunitha lodged a complaint with police, who registered cases under various sections of IPC, including 505 (1)B (intent to cause fear or alarm to the public) and 506(1) (threat and intimidation). Suresh and Ramesh were arrested and later remanded to custody.
Police had already warned of stern action against those spreading rumours about Jayalalithaa and formed special teams to investigate such cases.
Soon after 68-year-old Jayalalithaa was admitted to the Apollo Hospitals on September 22 after she complained of fever and dehydration, rumours were circulating about her health on social networking sites including Facebook and WhatsApp.