After they won their second-round match on Friday - but before any repercussions for what happened in singles had been announced - Fognini was asked about the possibility of being penalised.
"When someone makes a mistake, they apologise," he said. "Accepted or not accepted, at this point that's not up to me.
Fognini is married to 2015 US Open women's singles champion Flavia Pennetta.
"Everyone makes mistakes. Certainly, I've had bad days. And I will have others, like every human being."
Fognini and Bolelli won the Australian Open doubles title in 2015.
The board said the major offences were under the sections of Article IV, dealing with "aggravated behaviour" and "conduct contrary to the integrity of the game".
Violation of either section could lead to the permanent suspension from the four major tournaments and the $US250,000 penalty.
The board said there would be no further comment until the process is completed. Fognini, known as a volatile player, was fined $US27,500 ($NZ38,434), by Wimbledon in 2014 for his outbursts during a first-round victory.