The England Under-21 international has been heavily linked with a switch to the east London side, who are believed to have beaten off competition from Hull City.
Jenkinson initially appeared set to seal a permanent move away from the FA Cup winners after Arsene Wenger snapped up Calum Chambers in a £16m deal from Southampton.
However, West Ham manager Allardyce revealed that the Hammers are hoping to sign the right-back on a season-long loan in the next 24 hours.
“We’re interested in signing Carl Jenkinson on loan from Arsenal,” Allardyce told talkSPORT.
“I’m hoping it’s going to be completed in the next twenty four hours. We’re hoping it’s all going to be OK.”
Jenkinson has made 37 Premier League appearances for the Gunners since moving to the north London club from Charlton Athletic in 2011.
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