However, Allardyce insists that terms have not yet been agreed with the full-back and hinted that he has expanded his transfer network in an attempt to attract more players to the club following the £12m arrival of Jeffrey Schlupp.
"I think [Jenkinson's move is down to] negotiations between the chairman and the player's agent. While we appear to have an agreement with Arsenal, that agreement with the player hasn't reached a conclusion, so that's hanging in the balance, as we say," he told reporters.
"We're still searching for players. We've exhausted this country, we're more looking at the European element now, because of the vast cost of the players in this country. They're a really big price. Someone said to me 'You spent a lot of money on Schlupp'. It's not really; it's very little compared to what everybody else is spending and valuing their players at. He's actually quite cheap compared to what we've been quoted for other players. It's expensive on where we've been before but not expensive in the market.
"He brings a lot of energy, experience for such a young man. He experienced a Premier League-winning run last year. Can play in three positions - wide left, wing-back or left-back - that could be important in a small squad. He was desperate to come, desperate to leave Leicester even though they're champions."
Palace have also been linked with a move for Juventus full-back Patrice Evra this month.