The "holding hostage" phrasing implies that somehow the House needs to use the debt ceiling as leverage to cut spending. But to cut spending, the House just needs to not approve spending - no agreement by the Senate or President should be required. So there's something funny going on. I don't know what exactly - maybe they anticipate that the Senate will "hold hostage" spending on essential programs in order to get the House to approve spending on programs the House doesn't want to approve?
Anyway, the "holding hostage" phrasing would make more sense if the House were using the debt ceiling issue to try to force action on some unrelated issue (eg, abortion). Linking the debt ceiling increase with future spending cuts is not that sort of thing.