The Prime Minister stepped down in the wake of the referendum result in favour of the UK leaving the EU, after he had led the Remain campaign.
Mr Fox said that of the senior Conservative MPs considering standing to replace Mr Cameron, many were waiting for a leadership timetable to be decided by the party’s 1922 committee of backbench MPs.
“I'm not ruling it out, I'm not ruling it in - I'm going to think about it over this weekend," he told Sky News.
“I'll talk to my wife and talk to my colleagues and then you'll be one of the first few million to know.“
Mr Fox put himself forward for the Conservatives’ 2005 leadership election, which was ultimately won by Mr Cameron.
Since then however, Mr Fox was forced to resign his job as defence secretary after it emerged he had given a close friend, Adam Werritty, access to the Ministry of Defence, and that he had accompanied Mr Fox on at least 18 foreign business trips.
After Boris Johnson, Theresa May is widely regarded as the most likely candidate to replace Mr Cameron.
The Home Secretary kept a low-profile during the later stages of the EU referendum campaign and could emerge to halt the momentum Mr Johnson appears to have gained after helming the leave campaign.
At the time of writing, Mr Johnson leads the betting odds as the “Boris bandwagon” continues to gather pace. Ms May is the bookies’ second favourite, followed by Northamptonshire MP Andrea Leadsom, Michael Gove, new work and pensions secretary Stephen Crabb, and Mr Fox, who is in fifth place.
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