The vote is not to grant drilling permits, but would determine whether council is interested in establishing what oil and gas reserves are available.
Council was split on the previous 2015 offer, which was voted on in February. Regional strategy and policy committee chairman George Wood had to use a casting vote to win council support.
This time the matter has been elevated to council's governing body committee, meaning all 21 councillors will vote. Mayor Len Brown would have the casting vote.
Councillors were still divided this week. Arthur Anae, Chris Darby and Wayne Walker said they would vote against. Darby said New Zealand needs to "wake-up" to newer and safer sources of energy.
"The risks aren't just environmental. We need to stop relying on fossil fuels and start investing in long-term solutions we can sell to the world," he said.
On the other side, Bill Cashmore, Denise Krum and Penny Webster said they were in favour.
"I don't see any harm in checking to see what we have available, especially when the pressure is not that dangerous. Any offshore oil would have to be sucked out, it wouldn't spray," Cashmore said.
Brown strongly hinted he would not support it.
"In the area proposed in this block offer, off the west coast of Auckland, the risks of exploration appear to outweigh the benefits," he said.
"In particular, I'm concerned for the protection of the remaining Maui's dolphin population, for Kaipara Harbour, which is nursery to an estimated 95 per cent of West Coast snapper stock, for the shellfish gathering, shorebird habitats and the recreational assets of our beaches."
Greenpeace climate and energy campaigner, Steve Abel, said he would make a formal presentation at Thursday's meeting.
"It's completely appropriate for Auckland Council, and every council in the country, to oppose exploration drilling on the grounds of their commitment to serious action on climate change."
Abel said more than 8500 members of the public had emailed Brown urging him to oppose oil and gas exploration.
