Twitter Inc. is expected to field bids this week, and Marc Benioff has been building a case to Salesforce.com Inc. investors and others that his company should be the buyer, according to people familiar with the matter.

Mr. Benioff is looking to make a splashy acquisition that would secure for Salesforce a treasure trove of data as well as a prized consumer brand, according to the people.

Mr. Benioff, whose recent approach to Twitter set off the bidding process, sees the social-media pioneer as an “unpolished jewel” with untapped potential in advertising, e-commerce and other data-rich applications he regards as important to the cloud-software juggernaut’s next phase of growth, the people said.

But the brash CEO, who lost out to Microsoft Corp. in a bitter battle to buy LinkedIn Corp. this spring, faces formidable obstacles. Alphabet Inc.’s Google may bid also, the people said, while media giant Walt Disney Co. has been considering its own offer.

While Twitter could cost upward of $20 billion, or more than a third of Salesforce’s roughly $49 billion market value, it would be more bite-sized for Google, the search powerhouse whose parent sports a market value of more than $500 billion. Disney, meanwhile, has a market capitalization of almost $150 billion.