Having paid a club-record fee of £5.5million for one striker on transfer deadline day in Stephane Sessegnon from Sunderland, it is understood Albion pushed the boat out further for Anichebe.
Although undisclosed, the suggestion is Albion splashed out £6million to land the Nigerian forward following an eight-year spell with Everton.
During his time with the Toffees, Anichebe scored three goals in seven matches against Albion, so when the club came calling he sensed a natural attraction.
"I've scored a few goals against the club," said Anichebe, on the club's official website.
"There have been a few moments in my history that have happened against West Brom.
"I scored my first Premier League goal against West Brom; I scored here on New Year's Day two years ago - when my dad was in hospital.
"I seem to pop up with a goal against West Brom. They are key moments in my professional career so far, and because of that something pulled me towards the club.
"For me, having been at Everton for so long, which is my team, for me to leave there had to be a reason - some sort of connection.
"I felt a connection to come to West Brom, and as soon as I heard there was an interest, I knew I was coming here."
Sessegnon, meanwhile, is eager to get his Albion career up and running after completing an anxious move from the Black Cats.
Sessegnon signed a three-year contract just before the window closed at 11pm after flying in from Paris in the morning.
The 29-year-old had been en route to hooking up with his Benin team-mates for a World Cup qualifier with Rwanda in Porto Novo prior to being diverted to the midlands.
"I'm happy to have closed the deal because the day was too long," said Sessegnon.
Before heading back out to Paris, Sessegnon added: "West Brom play very good football and have good players so for me it's a great choice.
"The gaffer (Steve Clarke) said to me he wanted me because he wants to play more football, add more movement and create chances.
"I'm happy to sign for West Brom and I'm excited to start now."