The sites of these two new pools have not been determined, although an indicative business case has recommended an inner-west pool in the Whau area.
Public feedback on the budget expressed "serious concern" about access to the council's 24 pools and dissatisfaction with over-crowded pools and long wait times in peak periods, such as school holidays.
People were also concerned about this lack of access being a barrier to teaching kids to swim and potentially contributing to higher rates of drowning.
Water Safety New Zealand figures show 22 people drowned in Auckland in 2017, twice the number of 11 in 2016.
Water Safety chief executive Jonty Mills said swimming pools are integral to water safety education, saying the barriers of travel, time and cost are prohibitive to many people getting access and one of the reasons why aquatic education levels are in decline.
"Our concern is the correlation between this and long term drowning prevention. We support the council's initiative to open new facilities and from a Water Safety New Zealand perspective, the sooner this can happen the better for local communities," he said.
The council's 21 local boards have also been strongly supportive of more investment in swimming pools.
Goff was unavailable to discuss funding for swimming pools, but a spokesman said the budget provided a total of $151m in the Flatbush and inner-west pools.
The mayor has said the budget will deliver the city's biggest-ever investment to tackle big issues such as transport, housing and the environment. There is no money for a downtown stadium.
Goff said a stadium is unlikely to be built until the 2030s. Funding could be provided in the next 10-year budget, which runs into the 2030s, he said.
The budget, due to be debated and decided by councillors on Thursday, also contains funding for a raft of new smaller projects such as $1.4m to implement a smoke-free policy and $475,000 to the Auckland City Mission to improve its Hobson St property.
A number of proposals in the budget benefit Māori groups.
The Tūpuna Maunga Authority, which manages 14 maunga, or volcanic cones, is seeking an extra $23.5m for a programme to restore the tīhi (summits), reinstate open space areas, develop tracks and viewing platforms and replace exotic trees and weeds with native trees.
Ngāti Whātua Ōrākei Reserves Board is in line for additional funding of $14.3m and To Poari o Kaipātiki ki Kaipara, which oversees the Parakai Recreation Reserve, a further $11.5m to improve the reserve.
The Independent Māori Statutory Board, whose members sit on council committees but not the governing body, could receive a further $35m over the 10 years to improve Māori outcomes, taking its budget from $111m to $146m.
Board chairman David Taipari said improving Māori outcomes included things like identifying and scheduling sites of significance, helping marae, providing for employment and raising the Māori economy.
Goff's budget proposal also reveals that $55m of new funding and bringing forward $53m of existing budget for the America's Cup and Apec leaders conference in 2021 will come at a cost to other city projects.
The $1b budget for a 10-year, city-centre programme will be cut by $150m, including completion of a greenway on Victoria St running from Albert Park to Victoria Park.