The Hammers have already signed four players this summer and on Thursday announced that they had agreed a deal for Ecuador's World Cup star Enner Valencia.
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Arsenal acquired Mathieu Debuchy from Newcastle earlier this week and have also targeted a loan move for Atletico Madrid right-back Javi Manquillo, meaning Jenkinson will be allowed to leave on a short-term deal should Manquillo arrive at the Emirates Stadium.
The Irons are open to signing Jenkinson on a permanent deal but a short-term move appears more likely, given the player's age and potential.
Allardyce is keen to retain a British core at Upton Park and is also determined to drive the average age of the squad down, with the 22-year-old fitting the new-look recruitment profile at the club.
West Ham retain an interest in Manchester City right-back Micah Richards – who has fallen out of favour at the Premier League champions – while Guy Demel looks set to leave Upton Park.
Leicester have been linked with a move for the 33-year-old while his former club Hamburg are also monitoring his situation ahead of a possible summer bid.