Barnard worked for the Gunners in London and Singapore and was also at West Ham.
Apart for from being the Deputy DEO and commercial director at Ascot, she is also a non-executive director for Great British Racing.
Barnard graduated in Material Science at the University of Birmingham in 2004.
Arsenal’s interest follows Vinai Venkatesham confirming he will depart the club after 14 years next summer.
After arriving in 2010 as head of global partnerships, he worked his way up to CEO three years ago.
Venkatesham is also a non-executive director of the British Olympic Association.
He admitted quitting the Gunners was a “tough decision”.
But he is determined to oversee a “seamless transition”.
Arsenal director Josh Kroenke said the club’s board was “fully supportive of Vinai’s desire to pursue his next challenge”.
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But Kroenke added: “Change and succession is something the club is well prepared for.”
The Gunners have been on the rise under Venkatesham as CEO – with Mikel Arteta really finding his feet last season after taking over as manager in 2019.