There is speculation that some of the remaining candidates may still withdraw before Tuesday's vote - but that is unlikely to happen until they return to Westminster on Monday morning.
"Obviously we're happy with the results but there's still a long way to go to winning the contest", a spokesman for Johnson told reporters.
Three of the 10 candidates who polled less than the threshold of 17 votes on Thursday were eliminated: Mark Harper (10), Andrea Leadsom (11) and Esther McVey (9).
Assuming Johnson does not lose support, his tally is above the 105 votes required to secure a place in the last two.
In a joint statement, Jeremy Hunt, Michael Gove, Dominic Raab, Sajid Javid, Matt Hancock and Rory Stewart said the leadership contest was "a critical moment" for both the United Kingdom and the Conservative party.
The former mayor of London is the bookmakers' favorite to triumph in the race, but a series of debates and weeks of campaigning could cause a few twists in the tale.
Conservative legislators will hold further elimination rounds of votes next week until two contenders remain.
May, who remains prime minister, stepped down as party leader on Friday, having failed to deliver her plan for taking Britain out of the European Union after almost three years in the post.
Mr Stewart told the BBC's Politics Live he was "completely over the moon" to have got through the first vote.
But Thursday's voting revealed each candidate's current level of support.
The ballot takes place in a Houses of Parliament committee room between 10:00am (0900 GMT) and 12:00pm (1100 GMT), with the results expected to be announced around an hour later.
McVey is pursuing a no-deal Brexit, arguing that the agreement struck by May keeps Britain too closely tied to the European Union, while Hancock is against no-deal.
The second round next week will have a threshold of 33 votes, anyone polling less than that will be eliminated.
"We face a crucial choice: who can negotiate some better choices than the bad ones we face", he said. Four others, Dominic Raab, Sajid Javid, Matt Hancock and Rory Stewart also made it through to the second round next week.
International Development Secretary Rory Stewart, the contender most vehemently against leaving the European Union without a deal, also hopes to make it past the first hurdle.
After weeks of hustings around the country, the 160,000 grass-roots Conservative party members pick the victor, with the result announced in the week beginning July 22. May resigned as party leader earlier this month, and will depart as prime minister, after failing to sell lawmakers on an exit agreement.