Albion have been tracking Phillips for some time, but they have been unable to get a deal over the line.
Rangers were wanting £10 million for the 25-year-old, but they have reportedly lowered their asking price to attempt to make a sale.
West Brom are now willing to do a deal, as Pulis believes Phillips will enhance his attacking ranks.
Pulis is less convinced that Sandro would be a worthwhile recruit, though.
Sandro spent the back end of last season on loan at West Brom from QPR, and the Midlands club had the option to make the move for the central midfielder permanent.
However, it seems that Sandro did not do enough to convince Pulis that he would be a good signing.
Sandro is reportedly attracting the attentions of Benfica and Porto instead.