Browne is back with the Hammers and is keen to make an impression with his boyhood club having enjoyed a productive spell on loan at Oxford United last season.
With just a year remaining on his contract, however, there is plenty of interest in securing the 21-year-old midfielder.
The Hammers were impressed with Browne's performances at Oxford, where he scored eight goals in 42 appearances, and are willing to give him a chance.
Pellegrini's side have five friendlies ahead of the new season and it is expected that Browne will be given opportunities in those fixtures, with West Ham then ultimately making a decision over whether he'll be integrated into the first-team.
Fulham, meanwhile, are set for major upheaval after being relegated from the Premier League and they could lose Ryan Sessegnon, with Browne potentially seen as a replacement for the winger.
Season done. Very enjoyable one for me 💪🏽🔥. Time to rest up , recover and prepare for another big one ❤️⚒ pic.twitter.com/PnCz3K5YeR— Marcus Browne (@MarcusBrowne14) May 12, 2019
Rangers are in the market for more attacking talent too, while Celtic have reportedly issued a 'probe' after a document leaked online allegedly showed the club's transfer targets.
A host of Championship clubs were interested in Browne last January, while West Ham were also rumoured to be considering a recall. He did, however, stay with the League One club where he quickly became a star player.
Browne has played only once in the West Ham first team, coming on as a substitute in the Europa League fixture against Astra Giurgiu in the 2016-17 season.
The London-born midfielder has been with the club since he was eight years old and signed his first professional contract in November 2015.
West Ham see the potential of bringing through a youngster internally benefitting them in terms of their transfer budget and being able to strengthen other areas of the squad.