Succession planning at mutual fund giant Fidelity Investments calls for the next chairman to be Abigail Johnson -- daughter of current Chairman and Chief Executive Edward Johnson III -- people familiar with the matter said, according to The Wall Street Journal.

But she might not run the company as CEO, sources told the Journal.

Johnson has expressed doubts in recent months about his daughter's readiness to run Fidelity, say some of his confidants, according to the newspaper.

Under the plan he shared with Fidelity's board -- says a person familiar with the matter -- his daughter wouldn't necessarily become CEO at the same time she assumed the chairman's post, the Journal reported.

In one scenario Johnson described, Fidelity would have no CEO, and the heads of its operating units would report directly to the board, the newspaper said.

The Journal reported that another scenario envisioned a CEO who isn't a family member.

Fidelity spokeswoman Anne Crowley told the Journal: We have a succession plan that we reviewed with our board, but like most corporations, we're not going to make public our succession plans.

Representatives from Fidelity were not immediately available for comment.

(Reporting by Justin Grant; Editing by Lisa Von Ahn)