Widely regarded as one of the best players in the Championship, Wallace has been a man in demand this summer after opting to run down his contract with Millwall.
Burnley, QPR, Luton and Watford are all understood to have expressed an interest in signing the former Wolves man.
But it's Albion who are now set to win the race with Wallace's medical understood to be taking place today (Wednesday).
His signing will be a major coup for manager Steve Bruce who is believed to be a big fan of the right winger.
And family ties may have helped Bruce get the deal over the line – with Bruce’s son-in-law, Salford striker Matt Smith, good friends with Wallace.
Bruce said multiple times towards the end of last season that he wanted to refresh Albion's attack with the boss believing the Baggies lacked both pace and creativity.
The manager also said it was important to exploit the free agent market.
And Wallace's arrival follows the capture of John Swift – another free agent who Albion won the race to sign last month.