Myners who became interim chairman last May agreed to stay on until July of this year. However he is believed to be keen on taking on the role on a permanent basis.
The rift has appeared as Kevin Lomax, head of the nominations committee and a senior non-executive director is thought to want an outside appointment for the position.
It has been reported that Lomax believes that Myners has become too close to chief executive Stuart Rose.
The identity of a potential outside candidate has become less clear as both Baroness Hogg and Archie Norman are thought to have ruled themselves out while there is not thought to be a popular choice.