Jenkinson was a surprise omission from Hodgson's 23-man squad when it was unveiled this morning [Thursday] 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 Stokholm 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 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.
"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."
Hodgson's comments at least clarify Jenkinson's position, although the fact he has already chosen Glen Johnson, Kyle Walker, Leighton Baines and Ryan Bertrand for the full-back slots means there will be plenty of competition should the Arsenal man link up, as hoped, in Manchester on Sunday evening.
The England boss has also wasted no time in bringing Jack Wilshere back.
It is 17 months since Wilshere won the last of his five caps, against Switzerland at Wembley, and the 20-year-old has made only three appearances for Arsenal since returning from a catalogue of injuries.
However, with last weekend's dismissal at Manchester United meaning Wilshere must sit out Arsenal's Barclays Premier League encounter with Fulham at the Emirates on Saturday, Hodgson has opted to select the highly-rated midfield despite Arsene Wenger cautioning against such a move earlier this week.
As expected, the Three Lions will be skippered by Steven Gerrard, who will become only the sixth player in England history to reach 100 caps.
The other member of the 99-cap club, Ashley Cole, is amongst those who miss out through injury, with Chelsea team-mate Jack Rodwell and Manchester City trio Jack Rodwell, Joleon Lescott and James Milner also sidelined.
With Wayne Rooney and Joe Hart amongst the other senior players involved, it cannot be described as an experimental squad.
There are some youngsters though, including Liverpool's Raheem Sterling, whom Hodgson came across as a 15-year-old during his ill-fated stint at Anfield.
Sterling was drafted into the Three Lions squad for the World Cup qualifier with Ukraine in September.
However, that move was an emergency, 24 hours before the game.
This time Hodgson has made it clear he is in on merit, and not to ward off apparent interest from Jamaica.
"He's done extremely well for Liverpool," said Hodgson.
"He burst onto the scene at the start of the season and has been ever-present in recent games.
"It has nothing to do with him being courted by anybody else. My views on that subject are clear, as far as I'm concerned a player should have the right to choose who he wants to play for.
"But it would seem to me, having played at every level for England so far, it must be Raheem's hope that he's going to represent his country at national team level and he's on the threshold of doing that."
Sterling will be joined by Liverpool team-mate Jonjo Shelvey, who made his debut in the win over San Marino last month.
In addition, Celtic's Fraser Forster will be involved following his stellar performance in that sensational Champions League win over Barcelona at Parkhead last night.
The only completely new face is Leon Osman, who at 31, is finally rewarded for his consistency in Everton's surge up the Premier League table.
"He is a player I have admired since I came back to Fulham five years ago," said Hodgson.
"He's always done extremely well whenever I've played Everton or watched Everton.
"Of course, he's faced great competition in the position that he plays, but on this occasion we've got the opportunity and the space to bring Leon into the team."
Ashley Young, who has missed all England's qualification games so far this season after appearing in the August friendly win over Italy in Bern, is back following his recovery from a knee injury.
However, Gareth Barry, his one-time team-mate at Aston Villa, is overlooked.
Hodgson has opted to select just three orthodox strikers, with Danny Welbeck and Daniel Sturridge chosen along with Rooney, who became England's fifth highest goalscorer last month.