"I have asked councillors to stay at the table and to consider their options carefully. It's likely that the issue will be formally discussed at the governing body meeting Thursday," Mr Brown said.
Earlier this month, Mr Brown said he was with "most Aucklanders" on port expansion, saying there was an absolute limit on how far the port can go into the harbour.
One councillor told the Herald the council is likely to go with a compromise which the public will not like.
Another source said councillors were being primed to accept one of the two compromise options.
It is understood the ports company did not meet a request from Mr Brown on Friday for a formal response by yesterday to stop work on the extensions.
Ports chairman Graeme Hawkins has infuriated councillors by indicating the board has no intention of stopping work on the $22 million extension contract.
Stop Stealing Our Harbour spokesman Michael Goldwater said the council had an opportunity to do something significant for Auckland but the tail was wagging the dog.
"The port company has no intention of backing down. They are stalling for time due to the lack of leadership by Mr Brown and council," Mr Goldwater said.
Urban Auckland spokeswoman Julie Stout said the ports company was staring down the council, which was buckling with classic compromise options.
Urban Auckland, a society of architects and planning professionals, is challenging the lawfulness of the consents for the extensions in the High Court at Auckland on June 2.
Late yesterday, Devonport resident Carol Banks filed an appeal in the Court of Appeal in Wellington after the High Court at Auckland declined an interim injunction she brought earlier in the day to stop work on the wharf extensions.
Aucklanders split over port plan
Aucklanders are divided over plans to extend Bledisloe Wharf to accommodate bigger ships at Ports of Auckland, according to the latest Herald-DigiPoll survey.
The survey found 49.8 per cent of Aucklanders either opposed the extensions because they believed the port had sufficient land, or were opposed to the port being in Waitemata Harbour at all.
Another 45.9 per cent supported the extensions about 100m into the Waitemata Harbour.
Outside Auckland, 45.3 per cent of New Zealanders supported the port's cause and 35.4 per cent were opposed to the extensions.
When all the 750 respondents from Auckland and the rest of New Zealand were compiled, 45.5 per cent supported the port's position, 41 per cent were opposed and 13.5 per cent did not know or refused to answer.
The survey, conducted between April 17 and April 26, has a margin of error of 3.6 per cent.
There were 245 respondents from Auckland.
Ports spokesman Matt Ball said the poll showed most New Zealanders support the extensions, and their voice should be heard.
"The survey also shows that Aucklanders are evenly divided, they are not universally opposed to the port extensions as some have claimed," he said.