Jenkinson has spent two previous loan spells at West Ham where Allardyce was previously manager and looks set to join on a permanent deal after struggling to break through into the Arsenal team.
Speaking at a press conference, former England boss Allardyce said: "I think that's a negotiation between the chairman and the agent, but we appear to have an agreement with Arsenal [for Jenkinson]."
Jenkinson has endured a difficult season with the Gunners so far and was recently dropped for a Premier League match at West Ham after playing poorly in the EFL Cup exit to Southampton last month.
“He has lost confidence because the Jenkinson you see at the moment is the Jenkinson who can play, but is not completely Jenkinson because he has no confidence. That will come back. It is natural and very difficult", said Wenger.
“Jenkinson has not played for a long long period. You think you are there and you will only realise in six months that he is not completely there. And then you will play and realise you are not as good as you think you should be and you lose confidence. He is going through that period at the moment.
“Does he suffer in comparison to Hector Bellerin? Yes, because Bellerin is one of the best in his position certainly in Europe but Carl can compete with him. At the moment, no, because mentally he has not found back his total level and not his confidence.”
The boyhood Arsenal fan arrived at the club from Charlton Athletic in 2011 but has found himself down in the pecking order at right-back with Hector Bellerin and Mathieu Debuchy both ahead of him.