The voting, which began at 10 a.m., saw legislators arriving to cast their vote.
Punjab Chief Minister Amarinder Singh had called all Congress legislators in the state for breakfast before the voting.
The Congress legislators later gathered at the Punjab Bhawan here for a meeting on the issue.
Punjab has a Congress government since March. The party has 77 legislators in the 117-member assembly.
In the Haryana Assembly, ruling Bharatiya Janata Party (BJP) legislators Kamal Gupta and Latika Sharma were among the first voters.
The BJP has 47 legislators in the 90-member assembly.
The contest is between National Democratic Alliance (NDA) nominee Ram Nath Kovind and candidate of several major opposition parties Meira Kumar, who was former Lok Sabha Speaker.
The lawmakers, including elected MPs and member of Legislative Assemblies, are eligible to cast their vote to elect the successor of Pranab Mukherjee, who demits office on July 25.