The idea was discussed with the former PM, but Mr Cameron thought the job should go to a government minister.
It comes after Claire O'Neill, a former Conservative minister for energy and clean growth, was sacked from the role.
Former Foreign Secretary Lord Hague was also involved in discussions but will not be taking on the presidency either.
Downing Street has refused to comment.
News of Mrs O'Neill's replacement is expected to come in an imminent reshuffle of ministers.
Mr Cameron told the BBC it was "an honour" to be offered the job, but said a government minister should take the role because there would be "one line of command."
He added he has "a lot" of prior commitments he wants to carry on with, including his work as President of Alzheimer's Research UK.
The United Nations-led COP26 talks are the most important since the Paris Agreement to curb global warming was secured in 2015.
Countries are expected to deliver more ambitious domestic plans for cutting greenhouse gases by 2030, as current proposals are not enough to prevent dangerous temperature rises.
On Tuesday, Mr Johnson refused to answer questions at the launch of COP26 about who would replace Mrs O'Neill, who stood down as a Conservative MP at the election.
Downing Street told her she could not chair the meeting because she was no longer a minister.
Sources close to Mrs O'Neill say they think she was fired for criticising government failings.
She has accused the government of a "huge lack of leadership and engagement" on climate change and claimed the prime minister had admitted he "doesn't really get" the issue.
But senior cabinet minister Michael Gove told the BBC Mr Johnson was dedicated to environmental issues, and 30 years ago had described his political outlook as that of a "green Tory".