The Guardian says Woodward, 49, announced last month that he will step down from his role as the club's executive vice-chairman at the end of this year. He has been at United since 2005 and held that job since 2012.
Arnold, United's group managing director, head a list of three candidates to replace Woodward, all of whom currently work at the club.
Cliff Baty, the chief financial officer, and Hemen Tseayo, the head of corporate finance, are also reportedly being considered for the role by the club's owners.
Arnold currently reports to Woodward along with United's first director of football John Murtough and this power structure is likely to remain once Woodward departs.
Woodward announced that he will step down from his role on April 20 amid the fall-out from United signing up to the breakaway European Super League.