Mr Burnham said he thought the swimming initiative would be popular.
"It will be so obviously right, in that it improves the health of the nation, that it improves people's health, happiness, well-being, general quality of life," he said.
"If somebody is currently inactive, it's the most likely sport they're going to do. But unlike other sports, swimming has a barrier in its way and it's called an entry charge."
Shadow sports minister Hugh Robertson said: "If, as the government are saying, this is the Olympic legacy plan, today's announcement is about swimming for the over-60s - a great thing in itself - but that's hardly 'enabling young people through sport'."
Liberal Democrat culture, media and sport spokesman Don Foster said the government had "no chance" of getting two million more people involved in sport by 2012.
"There are lots of organisations across the country working very hard to try and do this, but it all should have started a lot earlier," he said.
"Ministers' fine words are all very well, but the figures for people dropping out of sport tell another story."
Olympic gold medallist Adrian Moorhouse welcomed the plans.
He said: "It's a great thing. I think you've got to start somewhere and swimming is a good sport to start with.
"You can't start with everybody straight away, but over-60s will really benefit."
Amateur Swimming Association chief executive David Sparkes said the money earmarked for the upkeep of pools would be a big boost.
"£60m will be a tremendous fillip in terms of refurbishing some of the pools we have. This will be a stimulus to local authorities to do that work.
"We also hope that some of this money will be used to build school pools to reverse the closures of these pools."
He said the number of public pools had remained constant at about 1,400 over the past 20 years, with the closure of older pools being balanced by the opening of new ones.
In addition, there is set to be an increase of Olympic-size 50m pools from 23 to 30 by 2012.
Help the Aged called the proposals a "positive commitment" but said not all older people would benefit.
The charity's head of healthy ageing Pamela Holmes said: "Free swimming for the over-60s will help those older people who like to swim, and local authorities will welcome the financial support for this programme.
Mike and Evelyn Ladin enjoy a free dip in the pool
"But for those people who don't enjoy swimming, or for those with problems such as poor strength, uncertain balance and weak bones, swimming classes offer nothing at all."
Olympics Minister Tessa Jowell also unveiled how the government plans to leave a legacy from the Games.
It includes promoting sport, regenerating east London, inspiring young people, and promoting business, tourism and jobs.
Schemes include encouraging walking and cycling, investing in construction skills and using the 2012 logo to recognise non-commercial projects inspired by the Games.
Ms Jowell said: "My ambition has always been that the Games will offer not just a great summer of sport, but the prize of changing people's lives for the better for generations to come.
"Today we are demonstrating how we will turn the rhetoric of the Olympic legacy into fact."
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