The CPI-M leader said support to Mukherjee did not mean backing for the Congress and that his party will "resolutely fight" the "anti-people" policies of the UPA government.
Forward Bloc General Secretary Debabrata Biswas said his party was not in favour of abstaining from the voting slated for July 19 for which the battle lines have been drawn between Mukherjee and P A Sangma.
RSP General Secretary T J Chandrachoodan said his party will abstain from the voting.
Bardhan said the Left parties will remain united though they are taking different stands on Presidential poll issue.
"As far as the Left Front is concerned, it will continue as ever, it will continue to be united as ever. This is only one single incident where there is a difference of opinion, a slight difference of opinion. We will continue our movements and struggles jointly," he said.
The CPI-M's decision to support Mukherjee came after a meeting of the party's polit bureau on Thursday morning in which the senior leaders deliberated on the issue.
Sources said one section of the party was for abstaining from the poll while another was for supporting Mukherjee. The section advocating abstention was of the view that they could not be seen voting for a Congress candidate. They said
Mukherjee had reached out to Karat on Tuesday seeking his party's support but the CPI-M leader had not given
any commitment. Mukherjee had telephoned Karat as part of the Congress exercise to get support from non-UPA parties.
Left parties account for around 51,000 vote value in the electoral college of 10.98 lakh vote value.
During the Left meeting, Forward Bloc backed Mukherjee, saying he has earned the "widest possible acceptance" and that it wanted to drive a "wedge" between the Congress and the Trinamool Congress. The party has two members of Parliament and 12 members of Legislative Assembly.
G Deverajan, national secretary of Forward Bloc, told PTI that the Left parties had earlier decided that the candidate should have the "widest possible acceptance" and in the present context, the Congress veteran has got support from the political spectrum.
"Mukherjee has support from all sides. The UPA is behind it. The NDA allies Shiv Sena and JD(U) are in his favour. Two non-UPA, non-NDA parties, BSP and SP are supporting him. Now he has the widest possible acceptance," Deverajan said.