Allan held talks with the Baggies on Friday after they agreed a fee with United for the highly-rated midfielder last month.
Dundee United were resigned to losing Allan after he rejected the offer of a new contract at Tannadice with his current deal expiring in the summer.
Old Firm duo Celtic and Rangers have also shown an interest in Allan, but it seems West Brom have won the race for the playmaker.
The 20-year-old will be in attendance for Saturday's FA Cup third-round clash against Cardiff at The Hawthorns to watch his prospective new team-mates in action.
Allan has also agreed personal terms and he is expected to seal the move to West Brom on Monday.
The Scotland Under 21 international's arrival will come as a boost to Roy Hodgson with Chris Brunt facing two months on the sidelines through injury.
