Alagiri, who visited Karunanidhi's ancestral house in Thirukuvalai near here, about 300 km from Chennai, however, ruled out floating his own party.
"If my followers ask me to contest from Tiruvarur in the by-election (whenever it is held), I will consider contesting," the Madurai-based Alagiri told reporters.
Karunanidhi died on August 7 after prolonged illness.
Besides Tiruvarur, Tiruparankundram assembly constituency in Madurai district is also vacant following the death of the sitting member.
Alagiri, who has of late been keen on re-joining the DMK from which he was expelled in 2014, said he was following the policies of his father.
He described as 'rumour' reports that BJP was behind him and that he had links with superstar Rajinikanth.
Alagiri, who has claimed that loyal party workers were with him, had organised a rally in Chennai earlier this month to press for his re-induction, but the DMK has been keeping mum on his overtures.