The Star Trek actor plays Sherlock Holmes in BBC drama Sherlock while Miller will assume the role of Conan Doyle's detective in new CBS series Elementary, with both shows transposing the character to the modern day.
Cumberbatch was initially reported as saying that he was "cynical" about the casting of his Frankenstein co-star as Holmes, before then claiming that his apparent criticism was a "misquote".
"It's a very high-class problem to have, to be over-scrutinised for every word you say," he told The Huffington Post. "But when people start quoting you for things you didn't say, that's when it becomes a bit weird and you lose control.
"I just made sure that [Jonny] was alright and of course he was, because he's a grown-up. I issued a statement saying, 'This is not what I said at all'."
Cumberbatch added that he is "such a fan" of Trainspotting actor Miller's acting work.
"The only shame about [the rivalry rumours] is that Jonny and I are friends, and they try to get a fight where there ain't gonna be none," he insisted.
Elementary premieres in the US on Thursday, September 27 and will air on Sky Living in the UK from October. Sherlock is expected to return to BBC One in 2013.
> 'Sherlock' vs. 'Elementary': How the US drama compares to the UK hit
Watch a promo for Elementary below: