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.
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.