The two broadcasters have been jointly covering the tournament since the 2014-15 season.
Both The Times and The Daily Telegraph are reporting that deal is now set to be extended through to the end of the 2020-21 season.
The FA Cup has provided excellent ratings for the BBC even when showing seemingly less attractive ties.
BT Sport is also believed to be delighted with the figures for its games.
Under the current deal both broadcasters show live games and highlights from the first round proper stage.
Each broadcaster shows one semi-final each with the final shown live by both.
In the early stages, the BBC also produces a Sunday "live goals" show featuring the action from eight matches as they unfold.
Reports suggest the new deal and its details could be confirmed later this week.