Three members of a family Bandhuprakash Pal, his wife Beauty and their 8-year-old son Angan were found killed in their house at Jiaganj in the Murshidabad district of West Bengal on Tuesday.
The police have detained four people in connection with the murders.
"It's a murder. The crime has to be solved. People want to know who are behind the killings. We want to know the same from the police. We are not interested in listening to speeches. We want a probe into it by a central agency," said Bengal BJP chief Dilip Ghosh.
Meanwhile, TMC Minister Ashish Banerjee visited the father of Beauty Pal and criticised the BJP for doing despicable politics over "such a tragic incident". "The BJP has come to a state that they may now be thinking of going to crematoriums and wait for corpses to call them their party members," said Banerjee.
Bengal BJP General Secretary Sayantan Basu alleged the TMC had colluded with "Bangladeshi miscreants" to drive out the original inhabitants of Jiaganj. "The administration has collapsed. The TMC along with miscreants from Bangladesh and the police are evicting the original inhabitants of Jiaganj," he said.
They were prepared to go to any length to prevent people from joining the BJP or the RSS. They feel this is the only way of retaining power in the state," said Basu.
The BJP and the Rashtriya Swayamsevak Sangh (RSS) have claimed that Bandhuprakash had been attending RSS weekly programme Milan over the past few months.
The police have, however, ruled out any political angle in the case and claimed the victims' family denied that Bandhuprakash had links with any political group.
The police said Bandhuprakash was in a serious financial crisis and a diary note suggested serious differences in the family.
The Criminal Investigation Department (CID) has now joined the investigation.
Terming the incident a "shame" for the people of the state, famed filmmaker Aparna Sen asked Chief Minister Mamata Banerjee to ensure that the perpetrators of crime were brought to justice.
The incident has also triggered a slugfest between Governor Jagdeep Dhankhar and the TMC. While the Governor on Thursday called the incident dastardly and barbaric, the TMC alleged Dhankhar was 'transgressing his constitutional jurisdiction'.
The governor in his statement had said the incident "shocked the conscience" on account of its enormity and it was "reflective of intolerance and worrisome law and order situation".
( With inputs from IANS )