This is a myth. There has never been such promise. Bush himself and all high ranking members of his cabinet deny it and recently even Gorbachev came out to put out the speculation and said there was never such promise.
http://www.brookings.edu/blogs/up-front/posts/2014/11/06-nato-no-promise-enlarge-gorbachev-pifer
We now have a very authoritative voice from Moscow confirming this understanding. Russia behind the Headlines has published an interview with Gorbachev, who was Soviet president during the discussions and treaty negotiations concerning German reunification. The interviewer asked why Gorbachev did not “insist that the promises made to you [Gorbachev]—particularly U.S. Secretary of State James Baker’s promise that NATO would not expand into the East—be legally encoded?” Gorbachev replied: “The topic of ‘NATO expansion’ was not discussed at all, and it wasn’t brought up in those years. … Another issue we brought up was discussed: making sure that NATO’s military structures would not advance and that additional armed forces would not be deployed on the territory of the then-GDR after German reunification. Baker’s statement was made in that context… Everything that could have been and needed to be done to solidify that political obligation was done. And fulfilled.”
The Brookings Institution is liberal, so evidently they would defend the notion of NATO expansion under German policy.