Digital radio station BBC 6 Music is one of those under threat
Union leaders have warned of industrial action after reports that the BBC is planning to close two national radio stations and scale back its website.
The broadcasting union Bectu and the National Union of Journalists (NUJ) accused the BBC of bowing to pressure from politicians and commercial rivals.
According to the reports, BBC Asian Network and 6 Music will be axed as part of the cost-cutting proposals.
The corporation said details of the strategy review had yet to be agreed.
Leaders of Bectu and the NUJ said they met a BBC director on Friday and were told the reports in the Times newspaper were "largely correct".
They expect to meet the BBC's director general Mark Thompson next week.
In addition to the reported closure of BBC Asian Network and 6 Music, the BBC is to reduce the number of its web pages by half and cut online staff by a quarter, the reports said.
The moves, according to the Times, are part of a strategic review which places quality ahead of quantity. Imported TV shows and sport are also set to be cut.
The proposals would reportedly yield savings of £600m, which could be redirected to higher quality programming.
Some commentators have said the BBC's plans were an appeasement to an anticipated Conservative government that believes the BBC should be cut down to size.
Gerry Morrissey, Bectu's general secretary, said the union would oppose the closures and claimed the BBC was "being bounced by its competitors and by the political climate ahead of the upcoming general election".
NUJ general secretary Jeremy Dear said the union would "fight the cuts with all its might".
"If true, these cuts will result not just in the loss of hundreds of jobs, but the loss of valuable, quality output aimed at young people and the Asian communities," he said.
"We will do everything that's required, including taking industrial action, if necessary, to defend jobs at the BBC."
The BBC said speculation was premature and the proposals would be presented to the BBC Trust in the near future. If approved, they are then expected to be put out for public consultation.
Despite the lack of confirmation over the fate of the digital radio stations, their supporters have spoken out against the leaked proposals.
Phill Jupitus, who was the first breakfast show host on the station, writes in Saturday's Guardian: "Cutting 6 Music is an act of cultural vandalism, and an affront to the memory of John Peel."
Sunny Hundal, editor of Asians in Media magazine, told BBC 5 live that cutting the Asian Network would deprive listeners of a valuable place where Asian content was integrated with mainstream output.
"There are a lot of people in Britain who produce hybrid British-Asian music and culture and they want to see some of that culture on display," he said.
David Elstein, former chief executive of Channel 5, said the corporation would face greater pressure after the general election, which is expected in May.
"I tend to regard this as trying to look smaller without ever actually being smaller," he told BBC Radio 4. "The BBC's income isn't actually going to go down.
"The bigger issue by far is that the BBC's income continues to rise every year. So the key issue that its critics have raised - which is the sheer size of the BBC - has not been addressed and I suspect that that will be addressed for the first time seriously after the election."