A BBC Trust review published today concluded that Radio 2 was highly regarded by audiences but should be more distinctive, while 6 Music needs to become more cost effective by growing its reach without increasing costs.
The review also suggests Radio 2 should have more speech at peak time.
"Radio 2 should provide the peak time audience with more content that the licence fee payer couldn't hear anywhere else," the review says.
"This meant, for example, refreshing comedy and arts programming and using some of this material in peak time."
6 Music on the other hand, which only reaches one per cent of the adult population, was compounded by low awareness, with just 20 per cent of the adult population aware that the station existed.
BBC Trustee David Liddiment, who led the review, said: "Radio 2 has a large audience who clearly love its output, but the review showed it must break out of the routine with its programming, in particular to be more distinctive during peak time. In contrast, 6 Music has a distinctive approach, but the review concluded that it needed to grow its audience base without losing its USP.
"We're aware of concerns about Radio 2 targeting a younger audience. The current average audience age of 50 is well within the station's target audience, but the Trust is clear that this must not fall any further, and we would like to see Radio 2 work on its appeal to over 65 year olds."
RadioCentre's Andrew Harrison, representing Commercial Radio, said: “We welcome the fact that the BBC Trust is calling for a greater contribution from Radio 2 to the delivery of the BBC’s public purposes, especially in peak times.
Radio 2 has a unique opportunity to offer licence fee payers the sort of distinctive output that can’t be heard anywhere else, due to its popularity and privileged funding position.
It is now down to the BBC to demonstrate that it can raise the bar and start to offer a more genuinely diverse and distinctive service across all parts of the schedule.
Comedy, arts, documentaries and specialist music are essential elements of Radio 2’s required output, but for too long it has been able to tuck these away at the margins of the schedule in its inexorable pursuit of popularity over public service.
We will be watching the fate of 6 Music very closely, but welcome the emphasis on this being more firmly positioned as an alternative station, with a greater depth and context to its output. At the moment it is suffering from a lack of clear identity.”