Are you sure they want the government to give the aid, or do they want aid to be given? If the latter, wouldn't they give freely of themselves, especially if they knew the government wouldn't give?
What do you think the American public's response would be, if Bush said "There is no Constitutional authority for us to send a single dime of taxpayer money overseas, but considering the enormity of the tragedy, aid is urgently needed, and I personally beg each of you who is able to give whatever you can afford to give, and to please do so as soon as possible." Do you think Americans (generally, not NYT reporters) would open their hearts and their wallets generously, or do you think they would call for his impeachement? Seriously.
You might want to read this: http://www.independent.org/publications/article.asp?id=1329
I can find no warrant for such an appropriation in the Constitution; and I do not believe that the power and duty of the General Government ought to be extended to the relief of individual suffering which is in no manner properly related to the public service or benefit. A prevalent tendency to disregard the limited mission of this power and duty should, I think, be steadily resisted, to the end that the lesson should be constantly enforced that, though the people support the Government, the Government should not support the people.5Cleveland went on to point out that “the friendliness and charity of our countrymen can always be relied on to relieve their fellow citizens in misfortune,” and indeed that “individual aid has to some extent already been extended to the sufferers mentioned in this bill.” Further, he suggested that if members of Congress really wanted to send seed to the suffering Texans, the congressmen might personally carry out this charitable transfer by using the seed routinely provided to all members for distribution to their constituents (at an expense of $100,000 in that fiscal year).