Writing in the early Tuesday edition of The Times newspaper, Osborne said he was not be the right person to lead the party because of his advocacy for the "Remain" vote, which lost in the referendum by 48 percent to 52 percent.
EU demands quick divorce from divided UK
"I fought the referendum campaign with everything I've got. I believed in the cause and fought hard for it," Osborne wrote.
"While I completely accept the result, I am not the person to provide the unity the party needs."
The race is on to find Cameron's successor as party leader -- expected to be appointed by September -- who would take over as prime minister but may call a general election to gain a public mandate.
Former London mayor Boris Johnson and interior minister Theresa May are considered to be the front-runners in the leadership race.
But support for May is higher than for Johnson, on 31 percent versus 24 percent in a YouGov poll, according to the Times.
FO says no immediate fallout of Brexit
Johnson was a leading campaigner for the "Leave" side. May backed "Remain" but did not play a prominent role in the campaign.
Health Secretary Jeremy Hunt, who called for the option of a second referendum on the terms of leaving the EU in the Daily Telegraph, may also be considering a bid.
"Britain must and will leave the EU. But we did not vote on the terms of our departure," Hunt wrote.
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ