The controversial campaign, devised by Hytner's predecessor Simon Gulliford, has had more than £40m poured into it since its launch two years ago.
Marketing has learned that Hytner, who quit his role as ITV marketing director to join the bank two months ago, has called for alternative creative strategies to be developed, and is expected to make a final decision on the ads' future early next year.
It is understood Hytner is in favour of axing the ads in favour of a simpler, back-to-basics strategy. However, a Barclays spokesman said "no firm decision had been made" and the company had not "ruled in or ruled out change".
The campaign, which included an execution in which Samuel L Jackson appeared talking about a chicken, has been criticised for being too obscure.
In research conducted last year by financial ad agency Masius on behalf of the Financial Services Forum, the ads generated high awareness, but only 18% of marketers surveyed thought it delivered a positive impact for the brand.