Last fall, Elliott Management, an activist investor known for buying small stakes and lobbing critiques at management, reached out to the chief executive at CDK Global and offered to buy the software company outright for $9 billion.

Around the same time in suburban Boston, EMC chief Joe Tucci met with executives from KKR. The private equity firm arrived not with its trademark buyout offer but with some news: It had quietly bought $250 million of EMC's stock and had suggestions for improving the business.