According to the Sun, the move could be completed by Tuesday, with Wilshere having already passed a medical ahead of his move across London.
The 26-year-old has agreed a three-year deal worth £15m, and will arrive on a free transfer having left Arsenal when his contract expired at the end of June.
Despite offering him a new contract, Arsenal could not tempt Wilshere into extending his long-term stay with the Gunners as he seeks regular first-team football in the Premier League.
The Sun report that West Ham wanted to get the Wilshere deal tied up on Friday, but 'a few minor issues' prevented them from confirming their fourth summer signing at that stage.
However, the issues - not related to his medical as some may have though - are set to be ironed out quite quickly, with Tuesday now looking to be the day that Wilshere will be confirmed as a Hammer.
Wilshere's family are all West Ham fans - which could have made his decision a much easier one - and they will be hoping that the former Gunner can go some way to replacing Manuel Lanzini, who has been ruled out for the entirety of the upcoming season with a serious knee injury.
West Ham have been trying hard to secure a new midfielder, although have recently missed out on Lazio star Felipe Anderson. Shock reports have also surfaced linking former midfielder Dimitri Payet with a return to the club.
Meanwhile, they have also bolstered their defensive ranks with the £20m signing of 21-year-old centre back Issa Diop from French side Toulouse.