Jenkinson was a surprise omission from Hodgson's 23-man squad when it was unveiled on Thursday morning as previously the England boss had stated his intention to call on the 20-year-old defender, who has impressed during early-season performances for the Gunners.
However it transpired the move relates to red-tape wrangling given the player's previous appearance at youth level for Finland and Hodgson hopes to rectify that prior to the Stockholm visit.
Hodgson told FA TV: "The reason I didn't include Carl from the start is because we haven't, at this moment in time, got clearance from FIFA but we do believe they are going to grant that clearance within a day or two.
"So by next Wednesday, he will be clear to play. He wants to play for England and has made that clear so we applied for permission to FIFA, as one is allowed to do, immediately after he left us and before we set off for Poland.
"That was a month ago but I didn't want to put his name down in a squad until such time as I've got FIFA clearance saying he's available to play for us in my hand. If the clearance comes through, we will put him in the squad. He was with us for training just before we left for Poland and I made it clear to him then that if he did want to play for England, I would include him in this squad and I intend to keep that promise."