Head coach Bryan Redpath has confirmed the move as part of each player's end-of-season review.
It comes less than 48 hours after Gloucester's Samoa international centre Eliota Fuimaono-Sapolu hit out on Twitter following the 12-10 Aviva Premiership play-off defeat against Saracens.
Fuimaono-Sapolu's comments were widely reported by the media.
It is possible he could now face disciplinary action from the Rugby Football Union, although no decision has yet been announced.
He tweeted: "I would say the referee was a joke but I'd probably get fined for it so I won't.
"Fair play Sarries. We score 20 more tries than them and the only try in the semi. Horribly boring but very clinical. Fair play.
"We scored two tries. One disallowed. No wonder my super15 friends are making jokes about northern hemisphere rugby. Boring rugby wins.
"The only semi-final that can't sell out!
"Farrell put more bombs on us than the U.S did on Osama Bin Laden."
Today, the 30-year-old's tweets included: "What happened to Journalism? It seems all they do is republish tweets and comment on them. Very deep."
The club, meanwhile, said on its official website today: "Gloucester Rugby head coach Bryan Redpath confirmed that he will be reminding all the members of the Gloucester Rugby playing department of their responsibilities with regard to posting on social network sites such as Facebook and Twitter.
"The club has never placed a ban on posting on such sites, instead taking the stance that such outlets fall under the domain of freedom of speech, but that the individual views of any employee do not necessarily reflect that of the organisation.
"Individuals are reminded, however, of their positions as public figures, of the fact that their opinions are open to public scrutiny and the possible consequences of that.
"A spate of postings on the Twitter network over the weekend has led to some media comment over the past 24 hours."
And Redpath added: "We are meeting with every player over the next two days as part of the end-of-season review process.
"As part of this process, I will remind each and every player of their position as a spokesman and an ambassador for Gloucester Rugby.
"Everyone is entitled to their own opinion, but, as professional rugby players, their views and opinions are monitored by thousands online.
"They have to be aware of this and responsible in their postings as a result."
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