Sky Sports revealed on Tuesday that the 22-year-old full-back was closing on a move to Upton Park.
With Arsenal having drafted in Mathieu Debuchy and Calum Chambers to cover the loss of Bacary Sagna to Manchester City at the end of his contract, Jenkinson remains some way down the Emirates Stadium pecking order.
A loan switch would allow him to get useful minutes under his belt, and there has been no shortage of Premier League interest.
West Ham have led the chase from the off, though, and Allardyce believes they are poised to complete another deal that shows the club are heading in an exciting new direction.
He told talkSPORT: “We’ve been interested in these last couple of days in signing Carl Jenkinson on loan from Arsenal and I hope it’s going to be completed in the next 24 hours, so we hope that’s going to be okay.
“I think we’ve gone out and tried to improve the strength and depth of the squad.
“We’ve tried to go a little younger, which I think is important today because they can be with the club for a long time.
“We have brought in young Cheikhou Kouyate from Anderlecht and Aaron Cresswell from Ipswich, who was the PFA left-back of the season for the Championship, so he’s a good young prospect.
“We’ve also brought in Mauro Zarate from Argentina, who scored 18 goals last season, so we have different dynamics in the front line.”