Hours after Microsoft made its $44.6 billion bid for Yahoo last Friday, Google CEO Eric Schmidt reportedly called Yahoo chief Jerry Yang and offered his help in fending off the software giant’s unwelcome advance. But how far can Google go in rescuing its rival before running into its own antitrust problems?

There are two ways Google could try to help the struggling search engine:

• Out-bid Microsoft and acquire Yahoo.
• Form a partnership with Yahoo.

The first option, of course, is a non-starter. Google (GOOG) may be able to get its DoubleClick acquisition past the competitiveness cops but it’s doubtful it could convince them that buying Yahoo is legit given its already dominant market share in online search and text ads. Plus, it’s not clear Yahoo wants to become a wholly owned subsidiary of what some techies see as the Evil Empire, Silicon Valley version. (Though answering to Mountain View is no doubt preferable to genuflecting before Redmond.)

Analysts say the second option, entering a commercial agreement with Yahoo (YHOO), is much more plausible, though even this could pose antitrust hurdles for Google. Here’s how a partnership between the two would probably work: Yahoo would outsource its search and advertising systems to Google (much like AOL does) in the hopes that cutting costs and bringing in more revenue per search query would raise its valuation and get investors excited about a Yahoo comeback, not a Microsoft (MSFT) acquisition.

Citigroup (C) analyst Mark Mahaney says there is a 25% chance that Google and Yahoo will partner (versus just a 5% probability that a white knight bidder will emerge, and a 60% likelihood that Microsoft will end up acquiring Yahoo). “We believe that the possibility of this outcome is likely greater than the financial markets believe,” Mahaney wrote in a report issued Wednesday. Mahaney also suggested that negotiations to outsource Yahoo’s European search operations to Google were already in the works before Microsoft made its $44.5 billion bid last Friday.

The upside of such a GoogHoo deal would be that, at least temporarily, Yahoo would remain free to compete in display ads, e-mail and instant messaging. But it’s not clear what the long-term implications of giving up its own search and ad platform and outsourcing to Google would be. It’s also unclear whether such a move would be sufficient to thwart Microsoft.

Most legal experts (including, of course, Microsoft’s general counsel Brad Smith) agree that the leading search engine couldn’t make an outright acquisition of Yahoo because of its already dominant market share in online search and text ads. but a growing number of antitrust specialists are now saying that even a commercial partnership between Google and Yahoo could pose legal problems.

An antitrust source close the the deal who did not want to be identified said a commercial agreement between Google and Yahoo would “really amount to agreements not to compete with each other” which is “generally said to be per se unlawful.”

“The question to answer is whether the arrangement would eliminate to a significant degree an independent competitor, in a situation where there is already a high level of concentration,” wrote Albert Foer, president of the American Antitrust Institute, a Washington, D.C.-based advocacy group, in an email.

However, a source close to Google who did not want to be identified says that the antitrust issues raised by a commercial deal between Google and Yahoo would “pale in comparison” to the anti-competition problems raised by the complete integration of Microsoft and Yahoo. Google’s chief legal officer last week questioned whether Microsoft would hurt innovation on the Internet if it acquired Yahoo and thus ended up controlling a large share of the e-mail and instant messaging market.

Of course, Google has monopoly over online search — the Mountain View, Calif.-based company has a 58.4% share of the U.S. market.

One thing is clear: Yahoo is running out of time, and will need to decide whether it will accept Microsoft’s bid or try to partner with Google soon. Yahoo’s annual shareholders meeting is in May, and Dinosaur Group analyst David Garrity says shareholders will want to see a decision made well before then.

Should Yahoo end up trying to enter into some kind of a partnership with Google, Garrity says you can bet Microsoft will “raise holy hell.”