Mr Diamond was promoted from head of Barclays' investment bank to oversee the group in 2010.Mr Agius had said previously that the FSA had not raised the Libor investigation when it approved his appointment.
In June Barclays was handed a £290 million fine for attempted fixing of Libor. Mr Agius, followed by Mr Diamond and two other directors, resigned soon after.
The FSA released minutes of a meeting between its chief executive Hector Sants and Mr Agius to the Treasury Select Committee of backbench MPs.
The document reported that at a meeting in September of 2010, Mr Sants "briefed" Mr Agius about the ongoing investigation by US and UK authorities into manipulation of Libor.
According to the minutes, Mr Sants said it agreed Mr Diamond's appointment "on the basis that the current view of the investigation does not have an adverse effect", but said it could change its position.
In a letter to committee chairman Andrew Tyrie, Mr Sants wrote: "I specifically made clear that we reserved the right to re-assess his suitability in the light of the conclusions reached by this investigation and requested they make this clear to Mr Diamond."
The file note also said Mr Sants told Mr Agius that the regulator's relationship with Mr Diamond had to improve.
It said: "HS [Mr Sants] explained that the relationship with BD [Mr Diamond] had not reached the level of openness, transparency and willingness to air issues with the FSA as is the case with John Varley [Barclays' outgoing chief executive]."
The note also recorded Mr Agius said "whilst BD [Mr Diamond] is very competitive ... he suspects he will now see him mature and relax given he has now achieved his goal" of becoming chief executive.
In his reply to Mr Sants, Mr Tyrie said information was "at variance with the impression we received from Mr Agius".
Barclays did not respond to a request for comment.