Terming the incident as a mere "scuffle", the ruling Trinamool, however, said the brawl was provoked by the CPI-M leader and he was now "resting" in hospital after creating a "drama".
Molla was admitted to a private hospital here with injuries after he was allegedly attacked by Trinamool members, led by former legislator Arabul Islam, at South 24 Parganas' Bhangar Sunday morning.
Following the incident, the Communist Party of India-Marxist (CPI-M) immediately issued a 48-hour ultimatum to the state government to arrest Islam.
"(They) assaulted Molla when he went to Bhangar to visit our party office, which was set on fire by Trinamool Congress workers last night," said CPI-M leader Surya Kanta Mishra.
"Molla got multiple injuries on his mouth and waist. He has been admitted to a hospital in Kolkata," Mishra said.
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Hospital sources said Molla's lip had been stitched and his condition was now stable. The Trinamool, however, rubbished claims of the CPI-M and said the party was trying to create disturbances in the state.
"Police advised Molla not to go to Bhangar, where the Trinamool Congress was holding a public rally. He forcibly tried to enter the area and abused our party leadership. That led to the scuffle," Trinamool leader and Urban Development Minister Firhad Hakim told a media conference.
Rejecting the CPI-M's demand for Islam's arrest, Hakim said: "If someone is arrested for a scuffle, then all the people present, including Rezzak Molla, should have been arrested."
"There has been a continuous process by the CPI-M to create disturbances at different places. Today's incident was part of that," he alleged.
The Congress and the Bharatiya Janata Party (BJP) also condemned the incident. West Bengal Pradesh Congress Committee chief Pradip Bhattacharya has demanded the arrest of the Trinamool leader.