His agent Martan Paledes is insistent a deal is done despite no official confirmation from the Hawthorns.
However Albion officials have indicated that the final details of the contract were being drawn up with an official announcement due tomorrow morning.
"This has gone through and we are very pleased," Paledes said.
"Borja is very happy and is looking forward to playing in the Premier League with West Bromwich Albion.
"We believe this is a great opportunity for him.
"Everything was sorted out this morning and he is very pleased to join the club."