The prison's top official has been asked to produce the print by December 8. The court has also asked for the Aadhaar identity card of Ms Jayalalithaa, who died a year ago while she was Chief Minister.
If the thumb prints don't match, as the opposition DMK has alleged in court, the results of three by-polls will be impacted.
In October last year, when Ms Jayalalithaa was in hospital, a doctor attested that she had an inflamed right hand and so had signed off on election affidavits of AIADMK candidates with a thumb print and not a signature.
The candidates were fighting by-polls in three constituencies. They submitted their documents with a thumb print of Ms Jayalalithaa.
Whether Ms Jayalalithaa was conscious when her sign-off was taken is part of an investigation ordered by the state earlier this year amid questions about the four-time Chief Minister's treatment and death.
She was admitted to Chennai's Apollo Hospital in September 2016 for breathlessness and she died on December 5.
P Saravanan, a DMK candidate who lost one of the by-polls to the AIADMK, challenged Ms Jayalalithaa's thumb impression and said it must be verified.
Mr Saravanan, a doctor in Madurai, said the print on the election papers "do not have clear ridges", indicating that the person was "either unconscious or dead". He also said rules were violated because an officer of the required ranking was not present at the time.
The court said it will compare the print with another that Ms Jayalalithaa had given most recently -- at the Bengaluru jail where she was spent 22 days after being convicted of amassing wealth disproportionate to her known sources of income. She was released on medical grounds.