In votes starting Thursday, the 313 Conservative lawmakers will narrow the field of 10 candidates down to two, who will be put to a vote of about 160,000 party members nationwide.
Worldwide development minister Rory Stewart has been holding conversations with the public over the past few days as part of his strategy, with former ministers Andrea Leadsom and Esther McVey among some of the other key contenders.
The final two will be put to the over 120,000 members of the wider Conservative Party in a vote from June 22, with the victor expected to be announced about four weeks later.
Like Mr Johnson, Mr Gove helped lead the campaign to leave the European Union, but scuttled Mr Johnson's bid to become prime minister in 2016 when he unexpectedly withdrew support and chose to run for the job himself - a move that gives him a lingering taint of treachery in the eyes of some Conservatives.
Hunt has said a very smart approach is needed to tackle the Brexit question, adding: "I will never take action to provoke a general election before we have delivered Brexit because that would be absolutely fatal for our (Conservative) party".More news: Scott Brown is co-leader at the RBC Canadian Open
Gove apologised but critics accused him of hypocrisy, noting that, in a previous role as education minister, he had signed off on rules to ban teachers for life for taking cocaine.
In the interview, Johnson also called himself the only one who could "beat the Brexit game" set up by Brexit Party head Nigel Farage and Labour leader Jeremy Corbyn, whom Johnson compared to Scylla and Charybdis, mythical sea monsters mentioned by Homer in Greek mythology.
He said he would seek a new Brexit deal and could not promise the exit would happen by October 31.
But he took that as a compliment while speaking to the BBC on Sunday.
He said on Monday that Britain had "suffered a loss of nerve" and must not delay its departure from the European Union beyond the October 31 deadline.
From the outsiders, Stewart has the most colourful past, having hiked through Afghanistan, smoked opium there, worked as governor of a province in Iraq, and acted as a private tutor to Britain's royal family.More news: Frozen 2 trailer shows Elsa, Anna on risky search for answers
The former Brexit Secretary, who held the post from July to November, resigned in opposition to the divorce deal that May struck with the EU.
For his part, Jean-Claude Piris, the former top lawyer of the EU Council in Brussels, summed up the mood in the EU capital when he told British newspaper the Guardian: "People in Brussels are fed up that the political class in the United Kingdom has gone a little bit crazy".
"The exit agreement was not negotiated against the British; negotiators on both sides tried, painstakingly, to find the best solution for all concerned", she told the Guardian.
Johnson, 54, has won the backing of many Brexiteer Conservatives by promising to lead Britain out of the European Union by October 31, with or without a divorce deal. But we do need clarity now.
Asked about "no deal", Hunt said: "I would be prepared to without a deal if there was a straight choice between no deal and no Brexit, but I'd do so with a heavy heart".More news: Bodies recovered as sunken Danube boat is raised in Hungary