But the Express & Star understands the German head coach was not the number one target for the club's board.
Instead, their preferred candidate was ex-Sheffield United boss Chris Wilder.
The 53-year-old enjoyed fantastic success while in charge of the Blades.
But he also clashed publicly with owner Prince Abdullah bin Musa'ad bin Abdulaziz Al Saud.
And that public spat ensured Albion owner Guochuan Lai vetoed a move for Wilder – despite the fact the Yorkshireman was keen on the role.
With Wilder out of the running, Albion conducted a thorough interview process which ended over the weekend.
And it was Wagner who is understood to have impressed them most with talks with German now at an advanced stage.
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The German boss famously guided Huddersfield into the Premier League back in the 2016/17 season.
He then defied the odds to keep them up the following year.
Since that achievement, though, the former Borussia Dortmund II head coach has seen his managerial career stall.
Despite his heroics at the club, Wagner left Huddersfield by mutual consent in January 2019 with the club sitting bottom of the Premier League.
In May 2019, he was appointed as manager of Schalke.
But he was sacked in September 2020 following a run that saw the Bundesliga side fail to win in 18 league matches.
Yesterday morning, you could find odds of 10/1 with bookmakers for Wagner taking charge at The Hawthorns.
But those odds have now been slashed to 1/10 on.
Both Wagner and the Baggies will now look to iron out any remaining details.
And if they are able to reach an agreement, an appointment could be made within the next 48 hours.