The Sunday Mirror says Bould was one of the key backroom team who, like Wenger, signed new three-year contracts last week.
The 51-year-old was chosen by Wenger to be his assistant two years ago when Pat Rice retired.
With three more years' experience in the No.2 role, it is hoped he will be ready to take over, should Wenger decide to retire in 2017.
"Steve and Arsene have a great working relationship," said a club source. "He communicates well and there is a lot respect for him."
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