The Baggies are known to want to sign Millwall's Jed Wallace and Leicester's Hamza Choudhury this summer.
The Express & Star understands Wallace in particular is a key target with the 28-year-old having become a free agent after opting to run down his contract at The Den.
Baggies boss Steve Bruce is understood to be a big fan of the former Wolves-man who primarily operates as a right winger but can also be deployed in the number 10 role.
Albion have always faced competition to land Wallace who is widely regarded as one of the Championships best players.
But according to reports, Burnley have now made a late move for the winger following the appointment of Vincent Kompany as boss.
Wallace has made no secret of his desire to play in the Premier League one day.
And it's highly likely he will sign for the club that he believes will give him the best chance of making it to the top flight.
Wallace got married just over a fortnight ago and has spent the past few weeks on his honeymoon.
The forward hinted he will make a decision about his future upon his return.
And that decision could be made within days with the vast majority of Championship clubs – including Albion – returning for pre-season training this week.
Alongside Wallace, fellow Baggies target Choudhury also has a host if clubs interested in signing him this summer.
The 24-year-old is expected to leave Leicester on loan during the transfer window.
Albion are believed to have been battling Middlesbrough for his signature.
But according to reports, Premier League sides Everton and Nottingham Forest are also keen on the central midfielder.
Leicester are understood to want 80 per cent of Choudhury's Premeir League wage to paid if they are to send him out on loan.
But that figure is likely to have to come down if Choudhury is to move to The Hawthorns.