Johnson's team has denied such tactics - but at least one backer suggested some MPs may have switched votes to end Gove's campaign, the BBC reported.
Thursday's vote put Johnson and Jeremy Hunt top in the race and they will now battle it out to become the next Prime Minister.
Hunt promised Johnson "the fight of his life" in the coming weeks, the report said.
Johnson's place in the final two had been widely expected as he had been the frontrunner in all four previous ballots of Conservative MPs, with Gove and Hunt vying for second spot.
Two ballots were held on Thursday, resulting in the elimination of Home Secretary Sajid Javid and Gove.
Johnson and Hunt will now have to convince the Conservative Party's 160,000 members to vote for them, with the contest ending in late July.
After Thursday's results, some of Gove's supporters claimed Johnson's backers may have voted for Hunt to eliminate their candidate.
However, Gove's campaign manager Mel Stride dismissed suggestions there had been a vote-switching operation, saying: "It doesn't seem to me on first observation of this that there has been.
"Because we didn't see a situation where, as some had speculated, a very large number of votes might have transferred from, say, Boris Johnson to Jeremy Hunt.
"It would appear to me everybody has behaved pretty much as one would hope they would."
Johnson and Hunt will now take part in hustings in front of Conservative Party members around the country, before members' postal votes are counted, with the final result to be announced during the week of July 22.
There will be a head-to-head debate on ITV on July 9, following previous leadership debates hosted by Channel 4 and the BBC.
( With inputs from IANS )