PR Sundaram, Lok Sabha MP representing this Namakkal constituency, made the remarks while addressing a public meeting here last night.
Sundaram, a former two-time MLA, also took a swipe at Jayalalithaa's arch rival and DMK President M Karunanidhi, who had originally raised the issue of the Chief Minister's health and asked her to take rest.
"The tongue of anyone who talks about Amma's health will be cut off," Sundaram threatened amidst loud cheers from the crowd.
The DMK chief had at a public meeting earlier this month said,; "Tamil Nadu is a state without a Chief Minister, she is not doing her job since she is not well and that is why she should take rest."
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The reference was to the AIADMK supremo skipping the Iftar event hosted by the party here on July 1 in which several prominent Muslims participated.
"I am unable to come to the Iftar party due to a sudden illness though it is my wish to participate in it (organised by AIADMK) as usual," Jayalalithaa had said in her address at the Iftar party which was read out by Finance Minister and party treasurer O Panneerselvam.
However, the Chief Minister had turned up at the Secretariat on July 16 to attend a series of official events.
Earlier today, Jayalalithaa chaired a meeting of her cabinet while also addressing her party's MPs ahead of the monsoon session of Parliament starting tomorrow.