An unnamed "top Republican congressional official" says that the economic stimulus bill could result in illegal aliens receiving checks from the government (link):

The legislation, which would send tax credits of $500 per worker and $1,000 per couple, expressly disqualifies nonresident aliens, but it would allow people who don't have Social Security numbers to be eligible for the checks.

Undocumented immigrants who are not eligible for a Social Security number can file tax returns with an alternative number. A House-passed version of the economic recovery bill and one making its way through the Senate would allow anyone with such a number, called an individual taxpayer identification number, to qualify for the tax credits.

As pointed out later in the article, the rules for the 2008 stimulus scheme were tightened up to require an SSN. ITIN numbers are, among other things, a way for the government to collect taxes from illegal aliens, and they've been used by banks to give home loans to illegal aliens.

UPDATE: The Associated Press report above is from Julie Hirschfeld Davis, and she's now updated the report to say the opposite: that illegal aliens won't be able to get those payments without a Social Security Number (link), quoting Harry Reid as saying:

"This legislation is directed toward people who are legal in our country. It is about time the Republicans got a different piece of reading material and get off this illegal immigrant stuff... This bill has nothing to do with anything illegal as far as immigration. It creates jobs for people who are lawfully in this country."

However, I'll leave this in the "unsure" category pending seeing the actual language being referred to. Illegal aliens can indeed file taxes using ITIN numbers, so unless there's language saying that only those who file using an SSN can get the payouts or some other restriction, then the original report would be the accurate one.

UPDATE 2: Media Matters for America cites chapter and verse, and it doesn't look like - at least under the House Engrossed version of HR 1 - those without Social Security Numbers can get those payments, which would exclude illegal aliens: mediamatters.org/items/200901300010