The FSA said it warned the bank two years ago that Diamond’s “suitability” might need to be “reassessed” as part of the LIBOR-rigging investigation which saw Barclays fined over £290m earlier this year.
The letters also reveal that there was a meeting on 15 September 2010 between the regulator's chief Hector Sants and Barclays' then chairman, Marcus Agius where these concerns were disclosed.
However Diamond had already been named at chief executive and formally took on the role in January 2011.
Sants also wrote to Treasury Select Committee chairman Andrew Tyrie disclosing his worries.
The letter said: "The FSA was fully aware that the ongoing investigation might come to conclusions which would be relevant to Mr Diamond's suitability.
"However, at the time, since the investigation was not concluded, it would not have been appropriate to prejudge its outcome.
"Secondly, I would like to record that in that conversation, I made clear that our concerns about Barclays' culture were not some generic observation but specific to Barclays, and asked that these concerns be communicated by Mr Agius to Mr Diamond. Mr Agius confirmed that he would do this."
The BBC reports that Agius had said that the FSA did not cast doubt on Mr Diamond's appointment during the meeting.
