Although the retailer has refused to confirm the appointment, Sky News on Friday announced that Swannell would succeed Sir Stuart Rose from next spring.
Swannell has spent the last 33 years as an investment banker. He joined Schroders in 1977 and stayed with the firm through its takeover in 2000 by US bank Citigroup.
In 2004, he helped defend M&S from a hostile bid mounted by Bhs owner Sir Philip Green, and also advised Roman Abramovich on his takeover of Chelsea Football Club.
The news of Swannell's recruitment marks the first senior-level appointment made by the retailer, since it revealed Marc Bolland as its new chief executive last November.