The Gunners full-back has impressed in the early stages of the season and the Three Lions manager is adamant he wants the former Charlton man in his side, but has to wait for permission due to the player's previous appearances for Finland at Under-21 level.
Jenkinson made the commitment to play for England last month and was included in a training squad before their games against Poland and San Marino.
Hodgson told the FA's official website: "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 that 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.
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"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."