A group of leading musicians has called in particular for a fair hearing to be given to members of a punk band facing prison for performing what they say was a "legitimate protest". The trio Pussy Riot face up to seven years behind bars after staging a performance in a Moscow cathedral calling on the Virgin Mary to remove President Putin from power.
Jarvis Cocker, Pete Townshend, Martha Wainwright and Neil Tennant were among those who raised concerns about the "preposterous" charge in a letter to The Times.
Last month's arrest of prominent opposition leader Alexei Navalny and around 500 others at one of a continued wave of anti-Putin protests following the election may also be raised. But Mr Cameron will tread carefully as he also seeks to exploit a relative thaw in relations between the two countries to push for stronger trade links.
Russia is already the UK's fastest-growing export market but the Prime Minister will argue that more needs to be done by Russia to persuade British firms to set up there.
Contacts had been broken off amid allegations that Russia was involved in the 2006 murder of dissident Alexander Litvinenko in London.