Shah targeted the TMC government, saying under it West Bengal had become the "most poverty stricken state" in the country, and promised to help it "flourish like Maharashtra and Gujarat" once the BJP came to power there.
Banerjee, who, after decimating the Left in its bastion in 2011, has been fiercely defending her turf, accused the BJP of trying to "intimidate" her party and declared she would "capture Delhi".
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The BJP chief who, with Prime Minister Narendra Modi, is being credited for the massive upswing in the party's fortunes, winning assembly elections in several states including Uttar Pradesh, said West Bengal was the "next" on his radar.
"Ebar Bangla (West Bengal this time)...I have seen great enthusiasm among the people during my visit and I am sure that the BJP will win (elections) in Bengal," he added.
"Only the BJP under (Narendra) Modi's leadership can provide such a government. No one else can," he claimed, adding, "Bengal will flourish like Maharashtra and Gujarat."
The mercurial TMC leader, who has been taking on the BJP over a variety of issues, including raids on her party leaders and their arrest in corruption cases, shot back, saying, "They (BJP) are afraid of the TMC. That is why they are trying to intimidate us.
"But the TMC cannot be browbeaten. We hold our heads high. Those who challenge me...I accept the challenge. We will capture Delhi," she said, addressing a public meeting in Birpara.
Apparently referring to Shah's visit to a slum in her Bhawanipur constituency in Kolkata yesterday, she said,"They come from Delhi and spread lies. They are in a hurry (to capture power in Bengal). They are threatening to unleash the CBI on TMC leaders. They cannot handle Gujarat, but they are eyeing Bengal."
Rejecting Shah's claim of an impoverished West Bengal, Banerjee said,"Development is the biggest pillar of our strength in Bengal. What we can achieve, nobody can match that. Our government works for the poor. No one can match the work done by us."
"India's industrial growth rate is 7 per cent, while that of Bengal is 10 per cent. India's growth rate in the service sector is 9 per cent, while that of Bengal is 13.99 per cent."
Banerjee took a dig at Shah, who had lunch in a Dalit household in Naxalbari a couple of days back, saying, "Lunch at a Dalit household and dinner at a five star hotel...We do not believe in such photo-ops."
Renewing her charge that the BJP incited people in the name of religion, she asked them to not align with the saffron party.
"We work for all the religions, castes, communities," she said, alleging that the BJP was "trying to destroy" West Bengal by its "divisive politics".
Shah, on the other hand, accused the Mamata Banerjee government of unleashing atrocities on the people in a way "never seen since Independence".
The BJP leaders have been accusing the state government of orchestrating "atrocities" on Hindus and appeasing the minority community.
With the political fortunes of Congress and the Left in a free fall, the BJP has been trying to claim their space in the eastern state.
Although BJP's vote share in West Bengal dipped since the 2014 Lok Sabha polls from 17.5 per cent to 10.2 per cent in the 2016 Assembly polls, the party won three seats for the first time contesting on its own in the state. Earlier it had polled around 4.06 per cent votes in 2011.
In the 2016 Assembly polls, BJP garnered around 56 lakh votes, up from 19.5 lakh in 2011. In the state where the party hardly had any presence a few years ago, its candidates had polled more than 10,000 votes in 262 out of 294 Assembly segments.