Phillips, part of Wales' squad for the November Tests, suffered the injury in Scarlets' loss to Ospreys on Saturday.
The 27-year-old, capped three times, was on the bench in Wales' historic victory against South Africa.
But Scarlets scrum-half Gareth Davies is fit again after a knee injury.
Once-capped Davies could face Ospreys in the Pro12 on Saturday after playing in Llanelli RFC's 58-21 Welsh Premiership win over Bridgend last weekend.
Davies was injured in Scarlets' 26-13 Pro12 win over Newport Gwent Dragons on Sunday, 5 October, ruling him out of Wales' November Tests.
The 24-year-old made his Wales debut as a replacement on their June, 2014 tour to South Africa having shone for the west Wales region last season.
Wales begin their Six Nations campaign at home to England on Friday, 6 February, before ending the tournament away to Italy on Saturday, 21 March.
Phillips was an unused replacement against the Springboks and Fiji during the autumn campaign and won his last international cap against Tonga in November 2013.
He will undergo surgery on his right shoulder on Friday, 2 January after being replaced just before full-time in the 17-15 defeat at the Liberty Stadium.
Fellow Scarlets hooker Ken Owens, who deputised for then-injured Richard Hibbard in South Africa, is also yet to return after undergoing neck surgery in September.
"Ken's still a few weeks away - we're not putting a deadline on that," said Scarlets head coach Wayne Pivac.
Gloucester's Hibbard was unable to play in Wales' November win against the Springboks because the game was played outside World Rugby's Test window.
That gave Ospreys hooker Scott Baldwin the chance to play in the 12-6 win.

The 28-year-old won in four hours three minutes nine seconds, more than eight minutes faster than Belgium's Pieter Heemeryck, with fellow Briton Mark Buckingham third.
In the women's race, Emma Pallant claimed victory from fellow Briton Lucy Charles, finishing in 4:35:15.
Switzerland's Daniela Ryf was third.
The race in the Canary Islands involved swimming 1.9km, cycling 90km and running a half-marathon (13.1 miles or 21.09km). The Ironman 70.3 event is half the regular Ironman distance.
The men's Olympic triathlon - which Brownlee won in 2012 and 2016 - involves a 1.5km swim, a 40km cycle and 10km run.

The former England paceman has been suffering from a "lower back issue", but is expected to feature in the five-match ODI series beginning next week.
Ireland play the Afghans in three T20s in India, staring on Wednesday.
"We're happy with the progress Boyd is making and feel confident that he will still make an impact on this tour," said Irish head coach John Bracewell.
"We are lucky that we have cover at present."
Bracewell added that bowler Tim Murtagh would be available for the ODI series following the birth of his second child.
The matches in Greater Noida conclude with an Intercontinental Cup encounter on 28-31 March.

A so-called Special Area of Conservation (SAC) has been created in the Inner Hebrides and Minches.
The facility is the largest conservation area for the marine species in Europe.
Scientists warned earlier this year that numbers of harbour porpoise were falling because of difficulties with catching food and breeding.
They had called on the Scottish government to act and introduce protected sites for the animals.
Environment Secretary Roseanna Cunningham said she was pleased to designate the first SAC and added: "This will help maintain the favourable conservation status and provide protection to the species and their habitats."
WWF Scotland director Lang Banks welcomed the move and said he was hopeful more such sites would be established in future.
He said: "It's fantastic news that Europe's largest Special Area of Conservation for harbour porpoise is to be located off the west coast of Scotland.
"We've long called for greater protection for this species and we look forward to working with the Scottish government and others to ensure that this site contributes to a coherent and well-managed network of marine protected areas."
Scarlets hooker Emyr Phillips is likely to miss Wales' Six Nations campaign as he prepares to undergo surgery on a dislocated shoulder, which will sideline him for up to 12 weeks.
[NEXT_CONCEPT]
Double Olympic champion Alistair Brownlee marked his debut in long-distance triathlons with victory in an Ironman 70.3 event in Gran Canaria.
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Ireland bowler Boyd Rankin has been ruled out of the T20 series against Afghanistan because of a back problem.
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A new conservation area has been set up in waters off the west of Scotland to help protect harbour porpoise.