Both acknowledged promoting a gaming operator on social media, the ITIA added.
The posts were removed immediately and the pair co-operated fully with the investigation.
The suspensions will only come into force if there is a further breach during the four-month period starting on November 11.
The ITIA said both were 'covered persons' under anti-corruption programme rules regarding directly or indirectly promoting or encouraging others to bet on tennis.
Former world number seven Fish won an Olympic silver medal as a player in 2004 and reached the quarter-finals of the Australian and US Opens as well as Wimbledon.
Bryan was a world number one doubles player with twin brother Mike, winning a total of 23 Grand Slams including mixed doubles titles.