The 25-year-old is one of a number of British & Irish Lions in Warren Gatland's Wales squad but still on the treatment table with Jonathan Davies, Gethin Jenkins and Jamie Roberts all among the walking wounded.
Davies remains hopeful however that Warburton is ready for Wales' opening Six Nations clash with Italy on February first, even if he's unlikely to play for the Blues before then.
"Sam had an injection last Saturday and the shoulder has settled well but he is seeing a specialist in London," Davies told Wales Online.
"He is improving all the time and is being assessed to see where he is. He is not available this weekend, but could be back in a week or two."
Davies also revealed that Jenkins' knee problem - picked up against Toulon - is the same injury he suffered when on last summer's Lions tour to Australia.
"It's the same injury he suffered against Australia (in November) but doesn't seem to be as bad," added Davies.
"He has had a scan and the feeling is it's weeks."