The opposition party has also asked the Election Commission (EC) to conduct a time-bound probe into the matter.
Congress leader Abhishek Manu Singhvi, speaking to the media, identified the deceased person as Mr Raju. The deceased was a BJP activist, killed in Mysuru in March 2016.
Mr Singhvi raised questions over the timing of the aid allegedly offered when Mr Shah, Union minister Ananth Kumar and Yeddyurappa visited Mr Raju's family last Friday, three days after the EC announced the Karnataka Assembly polls schedule.
The Congress submitted a memorandum to the EC, accusing the BJP of using money to influence votes in Karnataka, and said the ruling party at the Centre resorted to the "desperate measure, foreseeing its inevitable defeat in the May 12 Assembly polls.
The BJP has already said Mr Shah did not give any money to Raju's family.
Mr Shah and Mr Yeddyurappa, who has been given the title of No 1 Corrupt Leader' (by Shah himself), have blatantly and shamelessly violated the model code of conduct and committed the offence of influencing voters by giving a cheque of Rs 5 lakh to Raju's family with the sole intent of influencing votes of his family members, the memorandum submitted by the Singhvi-led delegation to the Commission reads.
EC must intervene and initiate action against Shah and Yeddyurappa for violating the Model Code of Conduct, it added.
The Congress had latched onto Mr Shah's comment at a press briefing on March 27 when he had mistakenly remarked that a Supreme Court retired judge has said that if a comparison is made on corruption, the then Yeddyurappa government would get the number one award.
Mr Singhvi pointed out that Mr Raju died two years ago and questioned the timing of the aid.
This is a very strange and suspicious event. Is this compassion and concern suddenly awakened because the EC has notified election in Karnataka on 27th of Marc, he asked after meeting Chief Election Commissioner OP Rawat here.
Mr Singhvi alleged the entire episode had cut at the root of the level playing field concept in the model code of conduct.
He claimed that it infringed a direct injunction against all forms of bribery and illegal gratification.
It brings the entire process into a non-equal system, which is quite against every rule of law, Mr Singhvi said.
These are very serious issues. We have, therefore, in conclusion asked for an immediate enquiry in shortest possible notice in days and not in weeks. We want the EC to disallow and prohibit repetition of such activities in Karnataka till May 12, he said.
Mr Singhvi said that the CEC heard the delegation patiently and has promised to look into the matter at the earliest in a time-bound manner .
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.