The Reds boss is coming under increased pressure following a poor start to the season.
Hodgson told Sky Sports News: "You read these ridiculous websites that we have and resignation has never once entered my thought and, what's more, never will.
"I pride myself on being a bit of a fighter, I pride myself on being able to stand the bad times.
"I've had a lot of good times recently - Barclays Manager of the Year last year, great accolades here and there and everywhere - I realise in football you don't get the accolades all the time, sometimes you have to bite the bullet and fine.
"It disappointments me that there are so many ridiculous websites, every day someone tells me that there is a new rumour about something that's supposed to happen and I think it's rather unfair that serious people use those rumours and then ask me questions because I shouldn't even be answering that question as it's literally never entered my mind and, what's more, won't enter my mind.
"I'm here for the long haul, I'm here to do a good job, I came believing I can help Liverpool rebuild and get going again, the players are very much behind everything we're doing."
He added: "So as far as I'm concerned it's work in progress, it's problematic at the moment because it's work in progress at the wrong end of the table, it's problematic because we're losing and not winning but there's only one thing to do with that - get out and play each week, play better, get the wins, get up the table and then you'll see my smiling face more often than my scowling one."