Google acquisition of Motorola Mobility is still warm and has only just been approved by the U.S. government, but already it sounds like the search giant is looking to unload the hardware division of the cell phone maker.

Last year, Google acquired Motorola to the tune of $12.5 billion, and that sale was only approved by the Justice Department in February. At the time, Google said it wouldn’t be favoring Motorola as a device maker over its Android partners, and it wasn’t long before Google revealed that probably the biggest driver for the sale was Motorola’s trove of some 17,000 mobile device patents.

Now, as Gizmodo reports, it appears as though Google is looking for someone to buy the hardware division of Motorola, which would confirm that what Google was really after were patents to use in the on-going war Apple is fighting with Android device makers all over the world. This would also appease other manufacturers of Android devices, as the perception is Google would of course favor Motorola.

The story cites a Wall Street Journal article that points to rumors that Google has offered to sell the hardware portion of Motorola to Chinese company Hauwei “at a high price.” There was some speculation that Google might get into the Android hardware business when it acquired Motorola, but the company has been pretty adamant about the fact that Motorola wouldn’t receive any favoritism over the many other Android partners Google maintains. Indeed, Android’s recipe for success has been in its number of available devices and device makers – Google probably would lose more than it could gain by turning Motorola into the Google handset maker.

And there are rumors that Google does intend to get into the tablet market with a co-branded device, but those stories claim Google is partnering with other device makers such as Asus to make the tablets. There hasn’t been much in the way of motion at all between Google and Motorola since the acquisition.So selling off the hardware division makes sense, in a lot of respects. Google doesn’t seem like it really wants to make hardware – it wants to sell advertising, which is what it has always done. Occasionally, Google steps in with a partner and makes a device, but even the Galaxy Nexus or Nexus One haven’t really exploded onto the scene. But ultimately, Google seems to be sticking to what it has been doing fairly successfully for the last few years: managing Android.

As Gizmodo and the WSJ point out, there is also the fact that Motorola’s hardware division hasn’t really been a spectacular performer of late, either. Motorola holds just 4 percent of the cell phone market, and the last five years in a row have seen losses for the company, to the tune of a total of $5.3 billion lost. If Google is trying to sell the hardware portion of Motorola, it would seem a potentially prudent decision.