Bush is asking for this authority, but it is unlikely to constrain spending. Read this (JSTOR) paper "Line-Item Veto: Where Is Thy Sting?". Excerpt: "Curiously, there exists little empirical support for the presumption that item-veto authority is important."
Or here is Robert Reischauer:
The crux of my message is that the item veto would have little effect on total spending and the deficit. I will buttress this conclusion by making three points. First, since the veto would apply only to discretionary spending, its potential usefulness in reducing the deficit or controlling spending is necessarily limited. Second, evidence from studies of the states’ use of the item veto indicates that it has not resulted in decreased spending; state governors have instead used it to shift states’ spending priorities. Third, a Presidential item veto would probably have little or no effect on overall discretionary spending, but it could substitute Presidential priorities for Congressional ones [TC: Hmm…].
Reischauer cites work by Douglas Holtz-Eakin:
Governors in 43 states [circa 1992] have the power to remove or reduce particular items that are enacted by state legislatures. The evidence from studies of the use of the item veto by the states, however, indicates no support for the assertion that it has been used to reduce state spending.
I have one simple model in mind: the legislature comes up with more individual pieces of pork in the first place. Can you think of others?