Morgan, 23, has started the first two league games of the season against Benetton Rugby and Isuzu Southern Kings.
However, the Wales international is now an injury doubt for the trip to Ireland this weekend after suffering a hamstring injury in training. Morgan will now be scanned by a specialist for the severity of the injury.
Head Coach Bernard Jackman said: “Tyler has a suspected hamstring strain after the backs units. He will get a scan today (Tuesday) so we are awaiting the results. Hopefully it is not too bad, but it’s highly likely he is out.
“We’re disappointed for him, but if it’s a low grade two then it is three or four weeks then he could be back mid-October. That is what we are hoping for.
“Adam Warren will come into the side in his place. We also have Jarryd Sage plus Connor (Edwards) and Tiaan (Loots) as options, but we will probably keep Connor and Tiaan with the A team.”
Dragons will also be missing fly half Gavin Henson with the number ten ruled out for a second successive weekend after a knee knock ruled him out of the Kings clash.
"Unfortunately it hasn't settled down the way we would have liked," added Jackman. "His knee keeps swelling up, after Benetton the swelling did not go down.
"Sometimes it blows up, you get the fluid out and it settles down but for him it's come back and he is seeing a specialist. We will know more when we get the report.
"It doesn't seem to be really serious but if we don't manage it properly now then it's one of those injuries that could become a constant issue for him. We can't afford to take any risks with someone like him. We are waiting for the best medical advice."
