BBC One controller Danny Cohen is planning changes in order to compete with ITV1 in the autumn, reports The Sun.
The show's hour-long episode may move from Fridays to Wednesdays, to face off against Coronation Street. ITV1 is moving its Thursday edition of the soap to Wednesdays as its Champions League coverage shifts to Tuesdays.
The One Show's move to Wednesdays is said to be due to being no EastEnders to compete against its rival soap on that night.
Chris Evans may receive a paycut if the move takes place, as he will not move to Wednesdays, and his Friday show will be halved.
A source said: "Moving the Friday episode to a Wednesday would see Chris's work cut in half so his pay would reflect that."
A BBC insider said that talks will be held with Evans if the changes were to take place.
Chris Evans welcomed the birth of his second child Eli earlier this month.