It is disputed in the sense only that there are contrary claims, but that's anything in politics.
It is '*not* disputed in the sense that the public record is in any way doubtful or ambiguous on the topic of whether Donald Trump is responsible for it. The 'birther' accusations date back at least to 2008 and the Democratic primary campaign, well before Donald Trump ever mentioned them, therefore under remotely no credible theory can he be accused of originating them. Even Factcheck.org, which is about as far from conservative sources as it can get, agrees with that.
However, on re-examining, the question over whether it was Hillary herself that put it forward in early 2008, or members of her campaign staff that put it forward, or independent Hillary-supporting PACs that put it forward without campaign authorization, is legitimately in dispute.