The Bhopal MP, who has been acquitted in the 2008 Malegaon blast case, moved the notice in the Lok Sabha Secretariat. Her colleague and party MP Nishikant Dubey had earlier sought privilege motion against Rahul Gandhi.
The counter-move from Dubey came at a time when Congress and other opposition leaders were seeking "unqualified" apology from Thakur over her remark on Godse.
The MP from Godda constituency in Jharkhand requested Lok Sabha Speaker Om Birla to allow him to move a privilege motion against Rahul for allegedly calling Thakur a "terrorist".
He urged Birla to ensure the dignity of MPs as custod of the House.
His demand came even as Congress MPs shouted "Down down Godse", "Mahatma Gandhi Ki Jai", "Down down BJP" during its protest against Thakur.
Dubey claimed the Congress has formed government with Shiv Sena in Maharashtra which in its editorial in the party mouthpiece 'Saamna' had hailed Godse as a patriot.
Thakur on Wednesday sparked a row with her remarks in Lok Sabha during DMK leader A. Raja's reference of a statement by Godse before a court on why he killed Mahatma Gandhi.
Facing protest by Congress, Thakur on Friday expressed regret and tendered her apology twice in the Lok Sabha over her Godse remark.
( With inputs from IANS )