Cox this week stated he might have to leave The Hawthorns for the sake of his career if he fails to hold down a first team place.
Cox has been thrust back into the spotlight given the back injury to fellow Republic of Ireland international Shane Long.
The 24-year-old made only his second Barclays Premier League start of the season in Tuesday's 1-0 defeat at Spurs, following that up with his trio of goals in today's 4-2 FA Cup third round triumph over Cardiff.
Hodgson initially expressed his delight with Cox's performance as he said: "That's a positive for Simon who has taken his chance when it has come along.
"I thought he did very well at Tottenham the other night as well, so with his three goals and his performance today, it's a clear positive for us and for him.
"The way he played today, he clearly showed he deserves a place in the team, and we'll be thinking very seriously about that.
"Simon's a good trainer, good finisher. In the finishing practices he is always right up there, so he has to be pleased with his day's work.
"I'm pleased for him, pleased he's shown he is a Premier League quality player and that he has taken his chance after being given it with great aplomb.
"On both occasions he has been amongst our best players."
But asked as to his thoughts when told of Cox's comments, Hodgson was clearly unimpressed, adding: "My thinking has never changed. I've never considered him going elsewhere.
"I don't know what he's said to the press, but as far as I'm concerned he is a West Bromwich Albion player, and I'm very happy about that.
"Before he makes those statements and gets you all so excited, it might be a good idea if he comes and speaks to me. It's the first I've heard of it."
A seemingly offside strike from Peter Odemwingie and Cox's first put Albion into a 2-0 lead after 33 minutes.
But against a Cardiff side showing 10 changes bearing in mind the first leg of the Carling Cup semi-final with fellow Championship outfit Crystal Palace on Tuesday, Albion surrendered their cushion.
Former Baggies striker Robert Earnshaw tapped home in the 36th minute and then substitute Simon Mason equalised five minutes after the break.
Cox, however, had the final say, initially with a thunderous 25-yard effort just after the hour and then a seven yard tap-in at the death.
Hodgson added: "It's good we didn't shoot ourselves in the foot by losing the game.
"We almost paid a high price after taking a sensible lead and than committing hara-kiri by giving the goals back.
"All credit to Cardiff they pressured us well, they didn't make it easy for us at any stage.
"But luckily we were able to get the third goal which put us on the front foot again, and after that we had a lot of chances to add to that, but we only scored once."
City boss Malky Mackay admitting to being "very proud" of his team given the number of changes he made.
"We managed to get back into the game and looked as if we were on the front foot, but it was a magnificent strike from Cox to take the win," added Mackay.
"We ran a Premier League side very close. At 2-2 I looked across at the opposition dugout and their manager was looking quite animated, so we were certainly doing something right.
"Now we can turn our attention to the game at Selhurst Park."