Sources said one section of the party is for abstaining from the poll while another is for supporting Mukherjee.
The section advocating abstention is of the view that they cannot be seen voting for a Congress candidate.
The meeting comes ahead of the Left party meeting in New Delhi to finalise its strategy on the issue.
Mukherjee has reached out to Karat seeking his party's support but did not get any commitment. Mukherjee telephoned Karat on Wednesday as part of the exercise to get support from non-UPA parties.
Sources said Karat was non-committal and told him that CPI-M [ Images ] Polit Bureau was meeting on Thursday followed by a meeting of Left parties where a decision on the issue will be taken.
Left parties account for around 51,000 vote value in the electoral college of 10.98 lakh vote value and Mukherjee has already garnered support of parties which has a total vote value of 5.44 lakh.
If all the votes are polled and counted valid, then the winning candidate needs to get atleast 5,49,442 votes.
The Left parties have kept a suspense on the issue of support to the UPA nominee, saying they were not bound to back the ruling coalition's candidate.
"We are not bound, we are not part of the alliance of Congress and UPA that we have to support anybody they propose," Karat had said.
The CPI-M had earlier decided to consult secular and Left parties before finalising a stand on supporting a Presidential candidate.
The Presidential poll, slated next month, was also discussed in the recent CPI-M Central Committee meeting, which authorised the Polit Bureau to formulate the party's strategy on the election after discussing the issue with other Left and like-minded secular parties.
Forward Bloc to urge Left parties to support Pranab
Meanwhile, Forward Bloc will ask other Left parties to back UPA Presidential candidate Pranab Mukherjee, saying he has earned the "widest possible acceptance", a move aimed at driving a "wedge" between Congress and Trinamool Congress [ Images ].
The party, which has two MPs and 12 MLAs, will try to persuade other Left parties to support Mukherjee in the meeting of the Left parties, G Deverajan, national secretary of Forward Bloc, told PTI.
He said 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. UPA is behind it. NDA allies Shiv Sena [ Images ] 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.
He said his party will strongly argue for supporting Mukherjee in the meeting.