The Scotland centre picked up a finger injury in the defeat to Ireland last weekend - with his pregnant wife, Annabel, posing a photo on social media of Jones with his arm in a sling.
Warriors head coach Franco Smith confirmed the player won't be available for tonight's home URC clash against Cardiff and will then be assessed on a weekly basis.
The South African said: "We're not sure how long Huw will be out for yet. It's something that can heal more rapidly than one thinks but he won't be available for the foreseeable future.
"He got injured during the match against Ireland, just before half-time apparently.
"It's his ring finger on his right hand.
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