Just speculation, but the animation studios probably take the initiative here the majority of the time. I'd say if there's a manga or novel that's very popular or has the potential to sell units, then the studio contacts the publisher of the series of interest and offers to do an animated adaptation of the work. Negotiations follow, and if the publishers and/or original author accept the terms and vice-versa, then the collaboration is set to begin and an animated adaptation is green-lighted.
I could be wrong about the original author having any influence over the decision though. Perhaps sometimes once an author is serialized the publishers gain complete control over any business decisions to be made with the property.
