Origi has failed to make an impact in Italy since moving to AC Milan from Liverpool last summer.
The Belgian striker is on a lucrative contract at the San Siro and AC Milan would like to move him before the window closes.
There have been little in the way of firm offers that Origi would accept however and the situation has been slow moving.
That could soon change though as Sheffield United, who have shown an interest in Origi along with West Ham, are pushing for him, according to Spanish outlet Relevo.
The Blades, who want to stay afloat in the Premier League this season, are seeing if a deal can be done with the Italians.
Origi enjoyed a productive spell at Liverpool and would be a Premier League proven option for Sheffield United.
Whether Sheffield United stepping up their interest in Origi spurs West Ham into action remains to be seen.
