Dell Inc. and private-equity firm Silver Lake are in advanced talks to buy EMC Corp., according to people familiar with the matter, a deal that would rank as the biggest technology-industry takeover ever and remove questions about EMC that have hung over the data-storage company for more than a year.

A deal for EMC could be reached within a week, according to some of the people, though it is also possible one won’t be reached. The deal currently being discussed calls for the buyers to spin off some of EMC’s VMware Inc. unit,...