The 25-year-old – who Wolves signed for £13million in 2017 – has been linked with a switch to Elland Road throughout the close season.
And the Express & Star understands the deal could be completed in the next 48 hours with Costa having said farewell to his Wolves team-mates at Compton on Friday.
The Portuguese international will initially join Marcelo Bielsa's side on a season-long loan deal. Leeds will then have the option to take him to Yorkshire on a permanent basis.
Costa shone for Wolves in the Championship while on loan from Benfica in the first half of the 2016/17 season. The deal was then made permanent in January 2017.
But the forward found game time hard to come by towards the end of last season as Nuno Espirito Santo’s side went on to clinch a seventh-placed finish in the Premier League.
Earlier today, Leeds completed a double loan signing with Manchester City midfielder Jack Harrison and Brighton defender Ben White joining on season-long deals.
Harrison, spent last season at Elland Road and scored four goals in 42 appearances for Bielsa's side. Leeds also have an option to sign the 22-year-old on a permanent deal at the end of the 2019-20 season.
White, who has signed a one-year contract extension at Brighton to extend his deal on the south coast until June 2022, has had previous loan spells at Newport and Peterborough.