Jenkinson struggled for first-team opportunities at Arsenal last season, with Bacary Sagna often preferred in the right-back role by manager Arsene Wenger.
Sagna has since departed for Manchester City, but with Mathieu Debuchy and Calum Chambers both arriving at the Emirates Stadium, Jenkinson's chances appear set to be limited once more.
On Wednesday, Allardyce explained that West Ham are in talks with Arsenal to bring the 22-year-old to Upton Park, as he looks to recruit young players to bolster their ranks ahead of the new Premier League season.
"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," he told talkSPORT.
"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 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 year, so we have different dynamics in the front-line."
