United are on the lookout for a new permanent manager to replace the sacked Ole Gunnar Solskjaer.
Ralf Rangnick will remain in interim charge until a new boss is appointed in the summer.
Mauricio Pochettino and Erik ten Hag are among the candidates linked with the Old Trafford vacancy.
Woodward was previously in charge of appointing and sacking managers at United
However, that will no longer be the case. Speaking on the Transfer Window podcast, Ian McGarry says Arnold has got his way in having sole responsibility for football decisions such as appointing a new manager.
Woodward will no longer have a "direct influence" despite not leaving the club for another two months.
McGarry said: "Richard Arnold's brief to find a new coach for next season has also been occasioned by the fact that Ed Woodward, who had been briefing that he would be consulting for the club on their recruitment of a new manager even though he's leaving his post, has now cleared his desk.
"And Richard Arnold was very keen that he should have sole charge of the administrative aspects of the football department. He seems to have got his way.