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Allies of Mr Sunak have contacted friends of the former Health Secretary, arguing that he does not have the support or 'infrastructure' and that they are competing for the same voters.
Mr Sunak stepped down after a major bust-up with Mr Johnson last weekend, with Mr Sunak allegedly having refused point black to cut corporation tax despite the PM's demands.
The scale of the fallout emerged in Survation research carried out on Wednesday - before the PM finally accepted that his time in power was coming to a close.
More than a dozen MPs are seriously mulling bids for the leadership after the PM's bombshell exit, with ministers alarmed that they will 'shred each other to pieces' to gain an advantage.
Culture Secretary Nadine Dorries wants to be a 'continuation Boris' candidate and believes she can be a disruptor, friends have told Mail+.