Maguire has caught the eye of the Hammers' coaching staff after impressing on trial at Upton Park.
The 18-year-old has been attracting the interest of a host of clubs after his displays with Waterford United over the past 12 months.
Maguire scored 12 goals in 24 starts for Waterford this term and he is regarded as one of the best young prospects to emerge from Ireland in recent years.
The likes of Blackpool, Brighton, Peterborough and Derby have all been credited with an interest, but West Ham are hoping to win the race to sign the Republic of Ireland Under-19 international.
The Hammers have now made a firm offer which Waterford are now considering with the Premier League club hoping to tie up a deal in time for the January transfer window.