The dolphin has disappeared for the moment and there are hopes that it has started to swim back out to sea.
Anyone who sees it is advised to contact the British Divers Marine Life Rescue service and to allow the creature to ground itself if it wants to.
If it does a rescue operation may be able to take place.
The organisation's director Fay Archell said: "It's incredibly rare and worrying for it to be that far inshore.
"It's a deep-water species which predominantly feeds on squid and they are not found in the Clyde, which is also comparatively shallow compared to what it would be used to.
"We hope that it has turned round and is on its way back to the coast. It's important that if anyone sees it they don't try to interfere with it or stop it."
Doreen Graham of the SSPCA said it was first spotted on Saturday further down the coast at Faslane.
"It was being monitored by MoD police and seemed to be fine," she said.
Little is known about Risso's dolphins which can grow up to 12ft long (3.6m).
They have stocky heads without a beak, are light grey to white, with a white chin and extensive scarring and are often seen surfacing slowly.
A spokesman for Clyde Coastguard said the creature should definitely not be so far up the river from the sea.
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