Mr Johnson is understood to be surprised by reports he had been urged to stand in the general election and has insisted he will serve out his term in City Hall.
Allegations of a rift flared up after reports on Saturday that Mr Osborne had personally delivered a message to Mr Johnson that he should fight in 2015, a move observers said was an attempt to call his bluff.
Some supporters of the mayor claimed the Chancellor's camp had embarked on a deliberate attempt to thwart any future bid to become Tory leader, according to a Sunday newspaper. It quotes an ally accusing Mr Osborne of trying to "destabilise Boris".
"This is nothing to do with party loyalty and everything to do with Osborne and Cameron trying to destabilise Boris.
"He saw it coming a mile off. They want to make sure that if the Tories lose, Boris gets as much flak as they do and can't stand as leader as a 'break with the failed Cameron-Osborne regime'."
It is understood there has been no conversation during which the Chancellor urged Mr Johnson to fight the election, but he is said to be relaxed about the reports. No offer has been made over the job of party chairman, but it is believed Mr Johnson would not accept such a proposal while he was running London.
Sources said discussions the Mayor has with Prime Minister David Cameron and Mr Osborne focus on the 2015 campaign, not his future role within the party.
Conservative party chairman Grant Shapps told BBC 1's Andrew Marr Show: "We all think Boris is a fantastic London Mayor and he's got a lot to offer.
"He's doing a brilliant job, his term runs to 2016, so he's got a little way to go. What Boris does is up to Boris, he's part of what we are trying to do as a government."