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Wolves are reportedly considering signing the mercurial striker on a short-term deal.
Costa, 33, is a free agent after terminating his contract with Brazilian side Atletico Mineiro in January.
The former Spain international is set to arrive at Wolves for a trial on Tuesday so that his fitness and suitability can be assessed, according to The Athletic.
Wolves are seeking attacking reinforcements after losing new signing Sasa Kalajdzic to injury just days after signing the striker in a £15m deal from Stuttgart.
Kalajdzic, who had been linked with a move to Manchester United in the summer, suffered an anterior cruciate ligament injury just 45 minutes into his debut for Wolves against Southampton on Saturday.
The injury leaves Bruno Lage with just one senior striker in his squad, Raul Jimenez, who was forced to withdraw from the Saints match after feeling pain in his knee in the warm-up.
Jimenez's injury is not thought to be serious and he is expected to resume full training next week.
Nevertheless, Wolves are keen to sign another forward as back-up and have now turned their attention to Costa.
Costa has a wealth of experience in the Premier League, having played three seasons for Chelsea earlier in his career.
After joining the Blues in the £32m move from Atletico Madrid in 2014, Costa went on to score 52 goals in 89 appearances for the west London club – winning two Premier League titles and the League Cup during his time at Stamford Bridge.
He returned to Atletico Madrid in 2018 and was part of the squad that won the La Liga title in the 2020-21 season.
In total, he scored 83 in 216 over the course of his two spells at the Spanish club.
In December 2020, Atletico agreed to terminate his contract, allowing the striker to sign a two-year deal with Atletico Mineiro.
But he left the club earlier this year after scoring just five goals in 19 appearances for the Brazilian side.
Fans have reacted positively to the news of the potential return of Costa.
"Oh my God yessss," wrote one fan on Twitter.
Another added: "I'm all for it, Costa back in the league."
Do you want to see Costa back in English football? Let us know your thoughts.