Mr Brown and Mr Joyce have clashed repeatedly over the rail loop.
The minister says an economic case for the 3.5km tunnel has not been made.
Mr Brown must provide funding for the rail loop in a 10-year budget being prepared under his name, and say where the money is coming from.
He said the project would require new funding sources if rates increases were to be held at around the rate of inflation
Details about possible tolls, congestion charges or a regional fuel tax will have to be released shortly if they are to be debated and included in the 10-year budget for approval next June.
Mr Brown's chief of staff, Phil Wilson, said: "There will be some reasonable concrete ideas in the long-term plan by June next year of alternative funding sources."
Mr Wilson said it would be another three or four years before tolls would be needed to finance the rail loop because there was enough money in existing budgets for preliminary work on the $2.4 billion project.
The council would need Government approval to proceed with new cash sources, such as tolls, he said.
Previously, the Council for Infrastructure Development has pushed for a $1 to $2 charge on vehicles using Auckland's motorway network to pay for big projects, and Transport Agency consultants believe tolls of $6 to $8 could pay for a new Auckland harbour crossing.
Speaking at yesterday's preliminary rundown of the 10-year budget to the council's strategy and finance committee, Mr Brown said the document would be a mix of principle with a strong pragmatic streak.
One of the "lines in the sand" for the mayor was no sale of council shares in Auckland Airport or Ports of Auckland.
Mr Brown appeared to drop an election promise to complete a rail link to the airport by 2020 and rail to the North Shore by 2025.
The mayor indicated there would no funding for the two big rail projects in the 2012-2022 budget, just work to protect the two rail routes.
Later, he said the two plans were still part of his vision, but he was mindful of keeping his eye on the inner city rail loop.
WAYS TO PAY
* 'Congestion charges' on roads into the city
* Road tolls
* New regional fuel tax
* 'Betterment charges' on properties