Johnson, a former United Kingdom foreign secretary, and Hunt, who holds that office now, are the last candidates standing after a series of elimination votes by Conservative lawmakers.
The victor of the runoff, due to be announced the week of July 22, will become Conservative leader and prime minister, replacing Theresa May.
While lambasted by left-liberal critics (and praised by grassroots conservatives) as a possible "British Trump" able to stand up to the European Union and deliver policies desired by voters despite opposition from the institutional left and big business, Mr Johnson is not the right-wing radical of popular imagination.
He will try to halt Johnson's momentum by picking away at his plans for Brexit as the two speak to meetings of party members across the country over the next few weeks.
The victor is expected to be announced in late July.
Following the result of the final ballot, Mr Gove congratulated his rivals and said he was "naturally disappointed but so proud of the campaign we ran".
Losing his Cabinet position after Bullingdon Club playmate David Cameron's resignation as Prime Minister, Osborne left Parliament to become editor at the Russian-owned London Evening Standard - among many other lucrative positions - where he has fought a rearguard action against Brexit and in favour of mass migration ever since.
A fifth ballot later today (local time) will determine the final two candidates.
He went on to praise Mr Gove as one of the "brightest stars in the Conservative team".
Johnson got 160 of the votes cast by 313 Tory lawmakers in the final round, and is strong favorite to win the ballot of some 160,000 party members.
They will also take part in a head-to-head debate on ITV on 9 July. "People should decide through a general election". As MPs started voting in the crucial fourth round of the contest, the favourite's outriders have been accused of plotting to lend votes to Sajid Javid to settle old scores with Mr Gove.