An interview with Pat Gelsinger from Intel reveals the chip giant was planning to top AMD's offer on ATI or go after NVIDIA but in the end they decided to do neither because of possible legal problems:
In an interview with The Inquirer, Pat Gelsinger, senior vice president and general manager of Intel's digital enterprise group, said Intel looked "pretty closely" at making a play for Nvidia or ATI, the two largest graphics chip companies in the world. Obviously, that never happened, as AMD closed its acquisition of ATI last year and Nvidia continues on as a standalone company.
Intel had some unique concerns that checked its ambitions, according to Gelsinger. "One issue was that we didn't know if we could because, if number one buys number two or three, what happens regulatory-wise?" Intel is the leading supplier of graphics technology for PCs because of its integrated graphics chipsets, and if it were to acquire a dominant share of the graphics market to augment its dominant share of the PC processor market, the U.S. government (well, perhaps the next administration) might have sat up and taken notice. And European regulators, currently hounding Intel on that continent, would almost assuredly have objected to the deal.