The 24-year-old is said to be the player that manager Glenn Roeder wants to bolster his midfield options and Brown would give him a more physical presence within the squad.
Brown moved to Bramall Lane in January 2000 in a £400,000 deal from Manchester City, and since his transfer he has established himself as one of the best midfielders in the division.
Manager of the cash-strapped South Yorkshire side Neil Warnock has already rejected a bid from Crystal Palace, but he admitted that a fee of around £3 million would be enough to lure Brown away.
"We don't want him to go, it's as simple as that," Warnock said earlier this month.
"In any case the bid is much less than we value him at, which is currently around £3 million, and if he was to go eventually it really ought to be to a Premiership club.
"We have kept Michael informed and he is delighted at the way things are going for him here. He signed a new contract not long ago, knows we want him to stay and is enjoying every minute of playing with Michael Tonge."
The Hammers are still active in the transfer market, but are waiting to see whether Paolo Di Canio's protracted move to Manchester United will be completed.
If Di Canio does move to Old Trafford this week then Roeder could very well make his move for Brown.
