Swansea boss Michael Laudrup says it is likely he will bring in a goalkeeper in January as Michel Vorm needs knee surgery.
Manager Jose Mourinho believes Ashley Cole will earn a contract extension at Chelsea.
A busy end to the transfer window saw the Nigeria forward leave Goodison Park along with Marouane Fellaini, who sealed a £27.5m deal to join Manchester United just before the 11pm deadline.
Anichebe's move became possible when Everton brought in Belgium striker Romelu Lukaku on loan from Chelsea for the season.
The Anichebe deal was a third of the day for the Baggies, who also signed keeper Lee Camp and broke their transfer record to bring in Sunderland forward Stephane Sessegnon.
They are reported to have spent £5m to sign the 25-year-old with clauses taking the possible cost up by a further £1m.
Anichebe has spent all of his career so far with Everton, making a total of 168 appearances across all competitions and scoring 26 goals, although many of his outings came off the bench.