When Airbnb was contemplating getting into the air travel business a year ago, CEO Brian Chesky considered launching an airline to reinvent the “actual experience of taking a plane,” according to a person familiar with the situation. CFO Laurence Tosi, a former Wall Street executive, thought it would be wiser to integrate with an existing flight-booking app, according to two people familiar with the matter.

The dueling ideas show the differences between Messrs. Chesky and Tosi, who have emerged as a key power duo amid Airbnb’s continued rise in Silicon Valley. More than two years in, Mr. Tosi has brought financial discipline and new focus areas—like luxury travel—to the company, which is preparing to go public as soon as 2019.

But like most relationships between founders and the experienced senior managers they bring in, the relationship isn’t always smooth. Mr. Tosi has been agitating for more power at the company, according to three people close to the company.