Bob Dudley, BP's chief executive, discussed the plan with Vladimir Putin last week at a meeting at the Russian president's retreat on the Black Sea, the newspaper reported, without citing any sources.
The appointment of a Russian director to BP's 15-strong board would be a first in the company's 104-year history, it added.
BP declined to comment.
Dudley and chairman Carl-Henric Svanberg put a proposal to buy a stake in Rosneft to Putin on Tuesday, Rosneft CEO Igor Sechin said on Friday, adding a new dimension to a long-running struggle for control of a big chunk of Russia's oil production.