The meeting will be the first formal deliberation of the top leaders of the ruling party a day after Mamata Banerjee and Mulayam Singh Yadav came out with Prime Minister Manmohan Singh's name as one of their choices in the presidential race.
The chiefs of Trinamool Congress and Samajwadi Party said they could also be agreeable on the candidature of former president APJ Abdul Kalam or former Lok Sabha speaker Somnath Chatterjee.
Mukherjee, who met Gandhi on Thursday morning, will be present at the meeting of the Congress core group which has senior leaders like AK Antony, P Chidambaram and Ahmed Patel, political secretary to Gandhi, as members.
Though some Congress sources ruled out the possibility of Singh participating in the presidential race, there was no official response so far from the party to the manner in which SP chief Mulayam Singh Yadav and TMC leader Mamata Banerjee mentioned the name of the prime minister.
Gandhi had on Wednesday informed TMC leader and West Bengal Chief Minister Mamata Banerjee that her party's first choice for Rashtrapati Bhavan was Finance Minister Pranab Mukherjee. When Mamata pressed for a second Congress choice, Gandhi mentioned Vice President Hamid Ansari's name.
A bigger surprise came immediately thereafter, when Mamata and Yadav disclosed the names of Kalam, Singh and Chatterjee as their preferred candidates.
This dramatic turn of events left the presidential race wide open and put the Congress in a dilemma. The party needs the support of both the TMC and the SP to ensure that its candidate wins.
Other allies such as the Dravida Munnetra Kazhagam, Nationalist Congress Party, Rashtriya Lok Dal and the National Conference, have already indicated their support for Mukherjee.