The proposed change in the law would be voted on in the Parliament in the New Year.
Downing Street had originally said that same sex civil marriage would be introduced at some stage before the next General Election and there was no mention of the proposed legislation in the last Queen's Speech.
But a senior source claimed that Cameron and Clegg have decided to go ahead with the approval of the law.
"The Prime Minister and the Deputy Prime Minister have agreed to get on with it. It was in danger of slipping beyond the General Election. David Cameron's view of this is 'get it done and get it done quickly'," The Telegraph quoted the source, as saying.
"If we are going to do this it's better to make it happen rather than have it hanging. If people want to commit to each other, he believes that's a foundation of a stable society," he added.
Cameron has promised a law to allow gay couples to marry in civil ceremonies, while not forcing the changes on the Church, the paper said.