The BBC Trust is the governing body of the BBC
"Public concern" might mean the BBC will have to "rethink" plans to axe two radio stations, BBC Trust chairman Sir Michael Lyons has said.
There has been growing opposition to the announcement by director general Mark Thompson on Tuesday, earmarking 6 Music and Asian Network for closure.
The proposals will go through a public consultation before being approved.
Union leaders have met with Mr Thompson, telling him the plans are "unnecessary and unacceptable".
In a joint statement, the BBC, broadcasting union Bectu and the National Union of Journalists, said that, at the meeting, Mr Thompson had "set out his vision for the BBC through to 2016".
"Both unions expressed their opposition to the planned closure of services and felt the proposals were unnecessary and unacceptable," the statement added.
A formal consultation will only begin once the BBC Trust has considered the proposals.
The unions, who say 600 people could lose their jobs, have already warned there could be staff strikes to fight any compulsory redundancies.
Former director general of the BBC Greg Dyke has told Richard Bacon's 5 live show that he does not believe 6 Music and the Asian Network will be closed down.
"All that's going to happen now is there's going to be campaigns to save the two stations they want to close," he said.
"I'm already being hassled by people in the street asking if I can save 6 Music. I keep saying I left them several years ago and it's nothing to do with me.
"There'll be a public consultation, and I suspect 6 Music will become enormously popular with all sorts of people writing in and I would take a bet today that it's still here in a year's time.
Greg Dyke talks to 5 live
"BBC 6 music doesn't get massive audiences, but the audiences who do listen to it are devoted and they'll make a big fuss."
Mr Dyke, who was in charge of the BBC when 6 Music was created and made the decision to make Asian Network a national station, refused to criticise Mr Thompson and the review.
"When I left the BBC one of the things I decided I wouldn't do was start criticising what my successor did," he said.
"It's always difficult for executives of any organisation to kill off the things they started but it doesn't necessarily mean it's not right to kill them."
However, Mr Dyke did admit that he would not have proposed the changes so close to a general election.
The proposed cuts have been criticised by music industry figures including music producer Mark Ronson and pop star La Roux.
More than 100,000 people have already joined a group set up on social networking site Facebook to save 6 Music.
Writing in the Guardian, Brit-winner Lily Allen said she hoped "that the backlash they've received so far will make them think twice".
The strategy review has also proposed that half of the BBC websites would close by 2013, along with teen services Switch and Blast.
But, Sir Michael has told the BBC's Today programme that nothing has been decided for definite yet.
"If we find that there's massive public concern that we need to take account of, then we will go back to the director general to rethink the strategy before it's approved."
Sir Michael said the report "heads towards a more disciplined and sharply-focused BBC" which means "some difficult choices" will need to be made.
"But we will not shrink from those choices where they are in the interests of licence fee-payers," he added.
"There will always be people who are disappointed when you take something away from them - that's the history of the BBC and the history of public services.
"One strong message that the trust is putting out is that the BBC simply cannot continue to grow forever. The impact that then places on both the licence fee and on competitors has to be more measured."
The review has also pledged that more money would be ploughed into international news and the BBC must put a higher proportion of the licence fee into quality programmes to make "fewer things better".