The Baggies are closing in on their first permanent signing of the summer after agreeing a deal rising to a total of £8million including add-ons for the 25-year-old.
To reach that figure, Zohore must make a set number of appearances, score a set number of goals, and help the club win promotion.
Albion have also approached Portuguese club Sporting over a deal for winger Matheus Pereira.
The Baggies are most likely looking at a loan deal for the 23-year-old Brazilian, who spent last season on loan at Bundesliga club Nürnberg.
But the Zohore deal is further down the line, and will be a welcome boost to Albion's threadbare front-line.
The Danish striker only netted once in the Premier League last season, but scored 21 goals across two Championship seasons before that.
Albion are in need of fresh faces up front following the sales of Jay Rodriguez and Salomon Rondon.
Zohore joined Cardiff in 2016 from KV Kortrijk, and was a key part of Neil Warnock's promotion-winning side in the 2017/18 season – scoring nine goals and registering five assists.
He struggled to replicate that form in the top flight last season, scoring just once in 19 appearances as the Welsh side failed to avoid relegation at the first attempt.
He's set to be the second signing of the summer folllowing Filip Krovinovic's loan move from Benfica, and the first permanent signing since Luke Dowling was made technical director last September.
Pereira is a Brazilian winger who moved to Portugal when he was young and came through Sporting's youth system.
He impressed for loan club Nürnberg last season, even though they were relegated, and was nominated for the Bundesliga Rookie of the Season award.
After selling Rodriguez, Rondon and Craig Dawson, the Baggies are now starting to bring in players.
New boss Slaven Bilic said it was 'crucial' the Baggies made additions this week and Zohore looks like he could be one of the first.
Albion also have offers in place for other players, as they look to rebuild their squad ahead of next season.
Allan Nyom, who spent last season on loan at Leganes, is expected to return to Spain.
The Baggies are in talks with Getafe over a deal. Nyom only has one year left on his contract and burned his bridges at Albion last summer.