The club have been forced to postpone Tuesday's training session, with the players instead having to undergo another round of PCR tests to ensure the virus has not spread among the squad. Ronald Koeman's news conference has also been called off.
Pending the results from the PCR tests, Barca say they hope to be able to reschedule training and the news conference for later in the day.
"After PCR tests carried out on Monday, two members of the football first team staff have returned positives for COVID-19," Barca said in a short statement.
"The club has informed the relevant sports and health authorities. Furthermore, the whole squad will undergo further PCR testing on Tuesday morning local time in line with La Liga protocol."
The positive cases work with the first-team squad daily, so Barca face an anxious wait ahead of Wednesday's trip to San Mames to see if any of their players have been infected.
Barca travel to Bilbao on the back of a six-game unbeaten run in La Liga following Sunday's narrow 1-0 win at bottom club Huesca on Sunday.
It's set to be the first game in charge for new Athletic coach Marcelino, who replaced the sacked Gaizka Garitano on Sunday.
Europe's top leagues have been increasingly affected by COVID-19 in the last few weeks.