"I think he has got to do it on the field of play," Hodgson said, according to The Mirror.
"With all the best will in the world, people can be in support or not in support, some will say he's not doing it, and some will say he's unlucky because he's not playing in the right position.
"But I don't have any truck with that. Joe is a midfield player, and what's the right position for a midfield player? Gerrard has played all over midfield for as long as I can remember.
"Joe has to come to terms with the fact that he has come here after an unsuccessful period at Chelsea with high hopes of taking Liverpool by storm and unfortunately it's not happened. That's life.
"He should have great confidence in his ability as he is a very good footballer. As far as I am concerned, everything he has to say should be said on that patch of green grass.
"He should be convincing me and everyone watching that this is a player who can help Liverpool win things. And I think you couldn't say that has been the case so far."
The Reds have a busy schedule through Christmas and New Year, and the manager now wants Cole to use the opportunity to prove himself.
"He'll get plenty of chances because we'll need everybody. I have not lost faith in him in any way. He is a good footballer. But you can't deny the start he wanted has not materialised," Hodgson added.
"I don't care how many things are written or said, you can't make black into white. But he is training well, it seems to me that his confidence is gradually returning and all Joe needs is a couple of good games or maybe a goal and that could turn things around for him.
"You'll have to ask him what isn't working. If he says, the only place he can play is behind the striker, he is limiting his chances of playing here. We also have Gerrard and Meireles who can play in that position, and we also have a situation where at the moment quite often we are playing with two strikers.
"I would be sad to hear him saying something like that, because he would be suggesting he can't do what he has done so often in the past for England and Chelsea, which is playing in a different position."
There had been suggestions that the ex-West Ham midfielder was not one of Hodgson's signings, but the boss maintained that he did play a part in bringing the player to Anfield.
However, Hodgson also insisted that Cole was not promised any specific position prior to his arrival.
He added: "I played my part in persuading him it was a good move, but I made it clear I couldn't promise him a certain position. I made it very clear Steven Gerrard was staying, and the competition for the one spot, if it was the one spot he wanted, it was going to be very tough.
"He's not so much a player I can really take responsibility for. I'd have to share the responsibility for Joe, but I have every faith in his ability, and I'm sure he'll come good."
