Mr Cameron fired the starting gun on the leadership race yesterday by announcing he would be handing over the reins by the time of the Tory conference in October.
Under party rules it is up to MPs to decide the final two candidates before they are put to the Conservative membership in a postal ballot.
The Times reports that supporters of Mr Cameron will do their utmost to stop Mr Johnson from taking the top job.
"I would find it all but impossible to have him as my leader," a senior Tory source told the paper.
“I don’t know if he can be prevented from making it to No 10 but many of us are willing to try.”
Although Ms May supported remaining in the EU, she adopted a low-key approach during the campaign, leading some to conclude that she was paving the way for a possible leadership run.
Meanwhile, leading Brexit campaigner Liam Fox said the decision on the succession timetable was up to the 1922 committee of backbench Tory MPs.
“The timetable for a leadership election in the Conservative Party isn’t determined by the leader, it’s determined by the 1922 committee,”