The former foreign secretary so far has the support of 84 MPs - of a total 238 MPs who have declared their preferred candidate - well ahead of his nearest challenger Jeremy Hunt, who is on 35.
Michael Gove is third with 33 MPs supporting him, followed by Dominic Raab (24), Sajid Javid (19), Matt Hancock (16), Mark Harper (eight), Rory Stewart (eight), Esther McVey (six) and Andrea Leadsom (five).
There are a total of 313 Tory MPs.
The result of the first ballot will be revealed on Thursday afternoon.
Candidates need to win 5% of votes (a total of 17 MPs) to move on from the first round.
If they do not meet the threshold, they will be eliminated.
Mr Johnson is widely expected to finish top - and the figures from the Sky News analysis will further enhance his status as the front-runner.
Many in Westminster see the contest as his to lose.
He formally launched his campaign on Wednesday, declaring that a third delay to Brexit would be catastrophic for the Tories.
"Kick the can and we kick the bucket," Mr Johnson said.
Mr Javid also launched his bid for the top job, declaring: "I believe now more than ever that this is a moment for a new kind of leadership and a new kind of leader."
Candidates needed to gain the support of at least eight MPs to stand, so contenders down the bottom of the list will have secured the backing of MPs who have yet to declare their voting intention.
Through a combination of phone calls, interviews and collating public statements of support, Sky News has been able to find out the intentions of more than 230 of the party's 313 MPs.
The Sky News analysis suggests that the battle to join Mr Johnson in the final two is between Mr Gove, the environment secretary, and foreign secretary Mr Hunt.
And it suggests that the likes of Mrs Leadsom, Ms McVey, Mr Stewart and Mr Harper could quickly fall by the wayside as the crowded field is whittled down.
The Tory leadership contest officially began on Monday, with 10 candidates vying to replace Theresa May.
After Thursday's first round, further rounds of voting will take place to whittle down the candidates to a final pair, with the 124,000 party members then asked to pick their preferred leader.
The final result is expected to be announced in the week beginning 22 July, with Mrs May then handing over power to the winner.