Walt Disney Co. Chief Executive Robert Iger will likely stay on past his 2019 retirement date if the entertainment company wins its bid to buy the entertainment assets of 21st Century Fox Inc., according to people familiar with the negotiations.

Representatives of 21st Century Fox and its executive chairman, Rupert Murdoch, requested as part of the potential sale that Mr. Iger stay on past his planned July 2019 retirement to assist with the integration of assets, valued today at $40 billion, and strategic repositioning of...