Hodgson reacted badly in the aftermath of Tuesday night's 0-0 draw with Ukraine when informed of the negative reaction back home, which Lineker had been partly responsible for.
In a series of Tweets, Lineker had condemned England's inability to retain possession, at one point describing the Three Lions' display as "awful", although he did delete that observation.
Hodgson said he was surprised by the negative criticism and now Lineker has outlined what he meant by his observations.
"Decent result from a team missing a few key players," said the second-highest England scorer of all-time.
"My only criticism, but it's a crucial one, was the inability to pass the ball accurately. Effort, defence, tactics, spirit, all sound.
"Roy Hodgson can't be blamed for some of England's finest having a bad night. With injured players returning his options will be greater.
"He may play Carrick as he's the only natural holding player we have, as well as our best passer. But who to leave out? Age old dilemma.
"Finally, I'd like to make it clear that I'm fully supportive of Roy and think he's doing a good job. Onwards.."
An argument between such a high-profile figure and the England coach would hardly be ideal given the importance of next month's encounters with Montenegro and Poland.
Yet Lineker's observations about ball retention are shared by many and, in fairness to Hodgson, pre-date his time in charge by some considerable margin.
With Paul Scholes now retired, England have no-one as consistently good at keeping the ball as Carrick but he was deemed surplus to requirements in Kiev as Hodgson opted instead for a three-man midfield consisting of skipper Steven Gerrard, Frank Lampard and Jack Wilshere.
The latter was one of those who underperformed on Tuesday night, although Hodgson insisted there was a legitimate reason for it.
"Jack is still looking for full fitness," said Hodgson.
"That is why we took him off in the second half.
"We certainly saw a much more effective Jack against Moldova, but he still did his work against Ukraine and, tactically, did all the things I asked of him."
The surprise was that when Hodgson opted to replace Wilshere, he did so with Ashley Young rather than Carrick, which triggered James Milner's switch into central midfield.
Ukraine, with a home game against a Poland side teetering on the brink of elimination and whom they have already beaten in Warsaw in this campaign, before a trip to San Marino, will feel the point they collected on Tuesday will earn them a play-off berth at worst.
What they will now hope - and England must avoid - is for Montenegro or Poland to spring a surprise when they head to Wembley.
"It was a game where neither side will think they have lost points," said Hodgson.
"Both teams showed they have got the quality to get to the World Cup in Brazil but the major danger, for us and them if we are going to be 100 per cent honest, came from set plays."
Hodgson might have been irritated at the negativity that greeted yet another draw in a group where they have still only beaten Moldova and San Marino, but it was warranted, if not for that one performance alone then what it came on the back of.
Ukraine are not a superpower, but then neither are England.
Shorn of Wayne Rooney's brilliance and the incision of Danny Welbeck, Rickie Lambert ended up isolated and had Daniel Sturridge not been forced to miss out with a thigh injury, the in-form Liverpool man would surely have been introduced.
As it is, the pressure Hodgson spoke of prior to kick-off will be turned up another notch over the next four weeks.
The manager though is keeping a sense of perspective.
"It's a very tight group and there are a number of teams who can still qualify," said Hodgson.
"We are hoping that it will be us.
"It only becomes a must-win scenario when you know if you don't win you cannot qualify. That would mean being outside the top two.
"If you are second in the group there are always the play-offs
"As far as I am concerned, that is where the must-win scenario will kick in.
"I am rather hoping that won't happen to us."
Despite this Lineker has pledged to continue being forthright in his views.
"I will continue to give my honest opinions about football," he wrote on Twitter on Thursday.
"I believe I'm entitled to. As are we all. You can agree or disagree. All good."
There has been much debate about England's style of football under Hodgson.
Everton manager Roberto Martinez believes any judgements should be reserved until after the qualifying campaign.
"There has been a lot of talk about this. In any qualifying group, assessing how the team has played comes at the end of that period," said the Spaniard, who has gained a reputation for flowing, possession football during his time in England with Swansea, Wigan and now the Toffees.
"Everyone wants to qualify. England have two home games now. It could be the point (in Ukraine) that allows England to qualify at the top of the group.
"I think we are getting carried away. I think we should be a bit more calm about assessing the performance of the team."
Surely the enthusiasm Sterling and Zaha would provide on the wings is what we are missing. The directness and the fear factor they would instill in other teams. Other teams just aren't afraid of us. But I reckon these 2 along with Sturridge and Rooney up front with Wilshire and Carrick in mid is our best bet with a bit more disciplined back line. I think it would mirror the way Liverpool are set up at the moment which is proving more than effective.- crazyballs