The two agencies are thought to have plans to move into premises near London Bridge, which would allow them to share back-office functions.
Both agencies are part of AMV, which is owned by Omnicom.
The move adds fuel to rumours that the two agencies are in discussions about a closer working relationship.
Several sources told PRWeek that talks between the two agencies range from a client-referral arrangement to a full merger.
One said: "Omnicom is looking to fold Fishburn into Fleishman. It makes sense to merge the two. Talks are quite far down the line."
However, management at both agencies denied any discussions were taking place.
Asked if the two agencies were in discussions, Jim Donaldson, managing director of FleishmanHillard UK, told PRWeek: "Not at this stage. There is no truth to the rumours about Fleishman."
Ali Gee, chief executive of Fishburn, said it was not true that the agency had "plans to work with Fleishman or merge".
Cilla Snowball, group chief executive of AMV BBDO, did not respond to requests for an interview.