The proposal has drawn sharp criticism from leaders of the Roman Catholic Church and the Church of England.
According to The Independent on Sunday, Blair has told friends that he "strongly supports the Prime Minister's proposal" to enact a new law to make gay marriage legal.
The Government announced the plan last September and a consultation is to be launched later this month.
Blair, who was prime minister from 1997 to 2007, is backing gay marriage despite strong reactions from his Church. Pope Benedict XVI on Friday urged the Catholic community to resists the "powerful political and cultural currents seeking to alter the legal definition of marriage".
Cardinal Keith O'Brien, the leader of the Catholic Church in Scotland, has also called the plan "madness", saying it would lead to a "grotesque subversion of a universally accepted human right".
The Archbishop of Canterbury, Dr Rowan Williams, and the Archbishop of York, Dr John Sentamu, have also expressed serious concerns.
While he was in office, Blair oversaw the introduction of laws on civil partnerships. However, he was not known to have spoken in support of same-sex marriage.
Equalities minister Lynne Featherstone believes marriage is the domain of the people and not the church. Cameron has said, "Society is stronger when we make vows to each other and support each other. So I don't support gay marriage despite being a Conservative; I support gay marriage because I'm a Conservative."
Archbishop Williams met the Pope on Saturday night, but it is not known if he discussed Cameron's plan to allow same-sex marriage.