Cricket Australia (CA) said they would wait until they received a report from Glamorgan to decide on a course of action for Marsh's recovery.
"Shaun's a tough guy, so it's certainly not a bruise the way he walked off holding his shoulder," Glamorgan coach Robert Croft said. "We'll have to wait and see.
"We hope the scan will be favourable."
Marsh scored two centuries for Australia in their Ashes victory over England and another two in a one-day international series defeat against the same opponents earlier this year.
Australia will be hoping Marsh can recover in time for the test tour against Pakistan in the United Arab Emirates, expected to take place in October, especially in the absence of banned batting trio Steve Smith, David Warner and Cameron Bancroft.
Frontline pacemen Pat Cummins (back), Josh Hazlewood (back) and Mitchell Starc (leg) are also racing against time to be fit for the Pakistan series.
(Reporting by Aditi Prakash in Bengaluru)
