Shane Duffy could be disciplined by Everton after a post appeared on his Twitter account expressing support for the IRA.
Duffy, 21, said his phone had been taken by a stranger before the message, which read "'Up The Ra" was retweeted in the early hours of Monday morning.
Everton manager Moyes is understood to be angry with the negative publicity generated and could opt to punish the County Londonderry-born defender, who is followed by around 48,000 people on Twitter.
The incident happened a day after the 20th anniversary of the IRA bombing of Warrington, and Duffy later tweeted an apology.
He posted: "cant leave ur phone anywhere now!!" and insisted: "The retweets and tweets was not me. Got to accept the hate. Really am sorry for this mess! Sorry guys!"
An Everton spokesman said the player had been on an evening out "when his phone was taken by someone who then retweeted an offensive remark, for which Shane has publicly apologised".