"You're asking for unconditional support and only my children get that. It's not something you give because you're under assessment the whole time," Cosslett told reporters.
"We're very pleased with Saturday's result but we keep a balanced view because we've had a disappointing run before that."
The Australian, whose contract runs till January 2021, will be asked to present his case in a post-tour review conducted by the RFU's chief executive Steve Brown.
England will face the Springboks again at Twickenham on Nov. 3, followed by matches against New Zealand, Japan and Australia.
(Reporting by Arnab Paul in Bengaluru; Editing by John O'Brien)
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