With Bacary Sagna out injured, Jenkinson had played every minute for the Gunners this season before Wednesday's League Cup tie against Coventry.
The 20-year-old has been part of an impressive defence that has conceded only two Premier League goals this season.
Hodgson names his squad for the World Cup qualifying double header against San Marino and Poland next week.
Finland are also desperate to call up Jenkinson and their strong interest could force Hodgson to make his move and claim him now.
He is eligible for Finland through his mother and has played for them at under-21 and under-19 level as well as England's under-17s.