The board appear keen to avoid a repeat of the early season troubles they suffered this season, with the new Hammers boss identifying Burnley's James Tarkowski as one of the targets he wants to seek to bring in over the summer window.
The club are also understood to looking to sell January signing Jordan Hugill, who managed to accumulate just 22 minutes of football since his £10m move to the club.
Michail Antonio and Winston Reid could also depart the London Stadium during the transfer window.
Former Premier League winning manager Pellegrini was appointed as David Moyes' successor earlier this week, and appears to be targeting defensive improvements with his early transfer targets.
The Chilean is understood to be keen to bring Burnley defender Tarkowski to the Hammers, as well as having a noted interest in Newcastle captain Jamaal Lascelles following both players' impressive seasons.
Tarkowski's form this season has seen his side secure European qualification for next season, as well as earning the player his England debut and a position on Gareth Southgate's World Cup standby list.
It's no surprise perhaps then that the player is gaining interest from fellow Premier League clubs, but Burnley and Sean Dyche are unlikely to let the key player go without a fight - and having last summer secured a high fee for Michael Keane, they know from experience they can push a hard bargain financially.
The signing of Tarkowski then would likely severely deplete the proposed West Ham kitty, but it would be the kind of high-quality signing that would be in-line with what Pellegrini is planning with West Ham.
The 64-year-old has vowed to inspire the kind of football that will light up the London Stadium next season – and to take the club into Europe.
Following the announcement of his appointment, he tweeted: “Proud and motivated to assume this new challenge at @WestHamUtd and to be back at @premierleague.
“We hope to achieve all the goals we set and take the club to international competitions.”