Reed, who took over the housing brief three months ago from Angela Rayner, said the government "always" expected "the big increase to come in the latter part of this parliament".
"I've heard it described as a hockey stick, you know, because it is a relatively gentle increase until it shoots up towards the end," he said.
Reed insisted the government had already "put in place huge changes" to increase housebuilding, such as the Planning and Infrastructure Bill, currently in its final stages in Parliament, which he said would "dramatically speed up" decision‑making.
A major part of the plan is to make it easier to approve homes near public transport, which Reed claimed "could be worth a million" homes on its own.
Earlier this month, the government also announced plans to shake up the house-buying system to speed up sales and save first-time buyers money.
The government's 1.5 million homes target is for England, as housing is devolved in Wales, Scotland and Northern Ireland.
The government's spending watchdog, the Office for Budget Responsibility (OBR), predicts 1.49 million net additional dwellings will be built across the UK during Labour's period in office.
The OBR says the supply of new housing is due to hit a low of 215,000 next year, but it predicts net additions rising sharply to 305,000 in 2029-30, before the next general election.
Just before last week's Budget, the Home Builders Federation (HBF) warned the government would have to go further to hit its target.
The organisation, which says its members are responsible for building 80% of new homes in England and Wales, said there needed to be more help for first time buyers while attacking a landfill-tax rise it says adds £14,000 to each build.