It's probable that there is some truth to Trump's claims.
It took quite a few hours until some reaction, and the reaction sounds like typical bullshit PR talk meant to distract from the full truth. The reaction sounds like the wiretapping accusations are actually mostly true, and that they are trying to keep Obama clean by resorting to a kind of "STRICTLY SPEAKING, the wire-tapping wasn't ultimately ordered by Obama HIMSELF, because STRICTLY SPEAKING, nobody in the white house has ever ordered the wire-tapping of any US citizen HIMSELF" argumentation.
Maybe some of you are not aware of this, but when it comes to problematic/immoral/illegal decisions, there are several strategies like "plausible deniability" etc. meant to prevent that the leaders themselves can be held responsible.
This is basically "Watergate 2.0", and the US administration has learned its lession from Watergate 1.0: Prevent that the president himself can be held responsible.
And of course, the very fact that Obama did not instantly deny these accusations himself, but rather let one of his puppies do the denying, speaks for itself.