Ipswich are also interested in signing Ameobi, who has netted 28 goals in 168 appearances at St James' Park.
But the 26-year-old, who has been dogged by injury problems, is understood to have his heart set on making the move to West Brom.
If the deal goes ahead, Ameobi will become Mowbray's first major capture of the summer.
A source close to Ameobi said: "Shola is 26 now so this is a big move for him and he knows a manager like Tony Mowbray will help to develop him as a player.
"He's over his injury problems now and he just wants to get his head down and play.
"He is keen to get his future sorted so ideally he'd like to get things done and dusted in the next week or so.
"A few clubs have shown an interest but he wants to go to West Brom."