The Baggies are keen on the 22-year-old Frenchman, who scored 28 goals for the Bees last season, but are behind Sheffield United and local rivals Aston Villa in the queue for his signature.
Albion will be able to significantly increase Maupay's salary, believed to be in the region of £8,000-a-week, but they cannot compete with the money the Premier League can offer.
The Blades have made Maupay one of their top strike targets of the summer, and Villa boss Dean Smith is also believed to be keen on his former player.
Villa are keen to sign another striker to back up £22m record signing Wesley Moraes, especially if Jonathan Kodjia leaves this summer.
Brentford are playing hardball over his fee, but United are the current front-runners in the race.
However, the Baggies are monitoring the situation in case Maupay's move to the Premier League falls through.
Albion were also keen on Preston North End striker Callum Robinson, but he has joined Sheffield United for £7million.
After losing Jay Rodriguez and Dwight Gayle, the search for a striker is Albion's priority, and how much they spend up top will have an impact on their business throughout the rest of the squad.
Between them, the Newcastle loanee and former Southampton man scored 45 goals in the Championship last season.
Although there was an understandable desire to somehow keep Gayle at the club, his £55,000-a-week Premier League wage packet is too much for Albion to commit to.
Rodriguez has been sold to Burnley for £10m after they triggered his release clause, giving the Baggies some money to reinvest in the front line.
Salomon Rondon is also expected to be sold this summer, although he will need to agree to move to China for Albion to get anywhere near his £16.5million release clause.
The Baggies are expected to sort their striking options first this summer before doing the majority of their business late on in the window.