Bachirou is struggling with a suspected broken toe - with the French midfielder in hospital for tests yesterday.
And defender Graham suffered a blow to his face which has aggravated the cheekbone he fractured during pre-season.
Moore told the Tele: "Both those boys are going to see specialists this afternoon so we will know something more after that.
"Wee Fred is struggling and we think it is the same toe he hurt before. We are sending him to go and see the orthopedist.
"And Andy's eye is still half closed after he got a blow on the face at the weekend, it's a concern as it's the same side he fractured before."
Bachirou said on his Twitter page: "Received a lot of messages today about if I was at the hospital today.
"Yes I was - for an X-ray."
Morton face Highland League side Deveronvale in the Scottish Cup third round on Saturday with a host of selection worries.
In addition to Bachirou and Graham, full-back Ross Forsyth has suffered a hamstring injury and is expected to be out for up to two weeks.
Loan goalkeeper Dominic Cervi is ineligible for the tie and fellow stoppers Colin Stewart and Matt McGinley are struggling with injuries.
Moore said: "It never rains but it pours. We can't play big Dominic on Saturday and Colin is also struggling with his hamstring.
"With Matty also out it looks like we might have to play Alisdair Deans and we have brought in a young boy from our under-19s whose been training with us."
However, the Ton boss also revealed that long-term injury victims Michael Tidser, Stuart McCaffrey and Marc Smyth all underwent a training session yesterday.
Tidser continues to battle back from knee surgery, while captain McCaffrey has a foot problem and Smyth has a hip injury.
Moore added: "We bleached the boys a bit with a bit of running yesterday and those three boys took part in some of it, not all of it, but some of it.
"I'm going to speak to Michael today to see if he's had any reaction to it and to see how he is feeling after doing it.
"But Stuart is looking like he could be out for another two weeks of so, although Marc Smyth will probably be back sooner."