Michael Matheson dismissed previous estimates the proposal - which is being championed by Scottish Labour - could cost £13.5 million.
Instead he told MSPs the costs could be much higher.
Mr Matheson said: “The reality is it is more likely to cost on an annual basis between £200 million and £230 million a year.”
Holyrood voted in favour of an appraisal, considering the costs and benefits of extending free bus travel to people under the age of 26, to be carried out as part of a review of concessionary travel for young people.
Labour, whose Scottish leader Richard Leonard announced the intention to bring in free bus passes at his party’s Dundee conference earlier this month, accused ministers of “parking” the proposal.
Transport spokesman Colin Smyth said: “Extending free bus travel will open up opportunities for young people - especially those on low incomes.
“But it is clear the SNP are in no rush to unlock the next generation’s potential.”
He added: “Instead of parking free bus travel for young people, the SNP should listen to Labour and use Scotland’s powers to transform our public transport system.”
Mr Smyth used the Holyrood debate to argue extending free bus travel to those under 25 was a “win-win”, which would help young people and also reduce the decline in bus travel.
The latest transport figures showed the total number of bus journeys across Scotland has fallen to 388 million a year.
While three-quarters of journeys on public transport are made by bus, Mr Smyth told how Labour wanted to “open up opportunities for Scotland’s young people”.
He said: “Transport costs are a huge burden on young people and their families, many young people earn below the adult minimum wage, never mind the living wage, and young people can find themselves spending half their income on travel alone.
“The cost of travel has become a barrier to opportunity and Parliament has the opportunity today to break down that barrier.
“The ability to pay should not determine young people’s access to education, to jobs, to social and leisure activities, but the reality is it does.
“Free bus travel will help put a stop to this injustice.”
He added: “It will provide young people with the same benefits it does for older adults and disabled people.
“But beyond that it will tackle the wider decline in our bus usage in Scotland, encouraging lifelong habits where the next generation choose public transport as their primary mode of transport.
“This policy is a win-win. It gives young people a break and invests in their future.
“And will help halt the dismantling of Scotland’s bus routes before our network disappears.”
Mr Matheson said: “Bus patronage has been in decline for decades now and the reasons for that are complex.
“The suggestion there is one simple solution to address that is misguided.”
He also stressed the Scottish Government invested more than £250 million for bus services and concessionary travel a year as part of its £1 billion funding for public transport.