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Woakes bowled two overs in the eight-wicket one-day-international win over Bangladesh before leaving the field.
Morgan said: "He'll have a scan tonight [Thursday] and we'll see what it comes up with. It doesn't feel right."
The captain added on TMS that man of the match Joe Root "showed his class, even [though he picked] up a niggle".
Root compiled a career-best ODI score of 133 not out at better than a run a ball, despite sustaining a calf strain.
Morgan, who scored an unbeaten 75 to help see his side home, described Root as "the glue in our side".
"He has scored a lot of runs in the last couple of years and continues to do it," the captain continued.
"He is not slow. He is batting at pace. He has been working on his power hitting and today it worked."
If Woakes was to be ruled out, England can still call up a replacement, with Middlesex pair Steven Finn or Toby Roland-Jones likely replacements.
Ben Stokes, who is recovering from a knee injury, bowled seven overs on Thursday and Morgan says the all-rounder came through it well.
"It was a stretch for him but he hasn't felt his knee all day," said Morgan.
Opener Jason Roy, under pressure coming into this tournament, received a vote of confidence from his skipper despite falling for just one.
Roy averages 35 at over a run a ball in one-day internationals, but since the start of May he has reached double figures just once in six innings and was dismissed flicking a delivery to backward square leg against Bangladesh.
"I thought Jason Roy was terribly unlucky today. Those things happen when you can't seem to score runs but we back him - he's been brilliant for us," Morgan said.
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England's Chris Woakes has been sent for a scan on a side strain and is a "worry" for the rest of the Champions Trophy, says captain Eoin Morgan.