Mr Hunt, now the longest-serving health secretary since the post was created in 1988, told ITV's This Morning that he was 'seriously considering' throwing his hat in the ring for a leadership bid.
A new Conservative leader is expected to be in post by the party conference in October after Mr Cameron announced his decision to step down following the victory for the 'leave' campaign in the UK referendum on membership of the EU.
Mr Hunt was among Conservatives who backed remaining in the EU, but has said voters would not choose someone who did not accept the referendum result.
Following the referendum result last week, the health secretary took to Twitter to call for 'calm and thoughtful leadership.
As remainer am in agreement with Dan Hannan just now: we voted to leave, now need calm, thoughtful leadership 2 bring &keep country together— Jeremy Hunt (@Jeremy_Hunt) June 24, 2016
Mr Hunt has long been tipped as a potential future Tory leader, but his chances were thought to have diminished as the junior doctors' dispute pushed him into bitter fighting with doctors and soaring NHS deficits left the health service increasingly in crisis.
Junior doctors have been voting since 17 June on whether to accept a new contract agreed in emergency talks that brought a wave of unprecedented strikes to a close. The BMA is set to announce results of the contract referendum on 6 July.
Meanwhile, Labour announced that Hackney and Stoke Newington MP Diane Abbott had been appointed shadow health secretary after the resignation of Heidi Alexander. Ms Abbott was a shadow public health minister until just under three years ago when she was sacked from the role by former Labour leader Ed Miliband.