Jurgen Klopp's side have taken just nine of the 18 points available in their opening six league games.
Wolves have also started slowly and earned their first win of the season as they edged Southampton 1-0 last time out.
But the midlands club saw summer signing Sasa Kalajdzic limp off with an ACL injury in the first half of his debut.
As a result, Wolves have looked into dipping into the free agent market to bolster their firepower - and have identified Diego Costa as a solution.
The former Chelsea and Atletico Madrid striker has been without a club since leaving Brazilian club Atlético Mineiro in January.
However, according to The Athletic, Costa's potential move to Molineux hangs in the balance.
It's reported that Costa has been denied a work permit to join Wolves.
He's said to be due to fly into the UK tonight before undergoing a medical on Wednesday.
Costa did not score enough points with his initial application but Wolves are appealing the decision.