The former Chelsea striker is still travelling to the club's Compton training base as Wolves wait for news of their appeal.
The 33-year-old is expected to undergo fitness tests ahead of signing a short-term contract, though the transfer is now at risk.
Costa has been a free agent since leaving Brazilian club Atletico Mineiro in January but his lack of action means he does not automatically qualify for a work permit.
Wolves turned to Costa after new £15million striker Sasa Kalajdzic suffered a serious knee injury on his debut in Saturday's 1-0 win over Southampton.
Costa scored 58 goals in 120 games for Chelsea between 2014 and 2017, winning the Premier League twice.
Wolves, who sit 14th, have netted just three times in six league games this season - although they have the best defence in the division having only conceded four.