Despite the closure of the summer transfer window last week, Costa is still eligible to make the move as he is still a free agent after leaving Brazilian side Atletico Mineiro in January.
Costa's inactivess this year saw him denied a work permit in the UK but after Wolves appealed the decision, their request has been granted to sign the former Spain international.
The 33-year-old will sign a contract until the end of the season, with Wolves currently short of options upfront.
New signing Sasa Kalajdzic suffered a knee injury on his debut against Southampton last week while Raul Jimenez is also of contention.
Costa previously spent three seasons in the Premier League with Chelsea, scoring 52 goals in 89 top flight appearances.
He won two titles with the Stamford Bridge outfit before a falling out with then-Chelsea boss Antonio Conte in 2017 led to his departure to Atletico Madrid.
Wolves will hope to have Costa available for this weekend's trip to Liverpool as they look to build on their six points accumulated from six games so far.
Costa was born in Brazil but has represented Spain at international level, winning 24 caps for La Roja and scoring ten goals.
