Rodgers led Liverpool to an unexpected runners-up finish in the Premier League last season but a failure to rediscover that form has seen the Reds slip to 11th in the table and suffer an early Champions League exit.
A mistake-laden 3-0 defeat at bitter rivals Manchester United on Sunday prompted media reports that the Liverpool manager had lost the confidence of his players.
"I hear one or two bits and pieces about unrest and it is totally untrue," Rodgers told a news conference on Tuesday. "Our dressing room is very strong.
"These are competitive players who nearly won the league last year, so they won't be happy with losing. But the actual spirit in the group is very strong.
"That is one of the reasons why, when we get that confidence back, then we will go into the second half of the season and be much better.
"We just need a little bit of luck and a break and I think we will get that. When we get some consistent results we can push on."
Liverpool have 21 points from 16 games in the Premier League, 18 adrift of leaders Chelsea. The pressure is mounting on Rodgers to get a positive result when Arsenal visit Anfield on Sunday.
The Northern Irishman, however, was not looking past the visit to championship side Bournemouth and will not rest players against lower league opponents with a place in the semi-finals at stake.
"This is the first cup competition that you can win, so it was always a priority for me. We wanted to do well in it," he said.
"At the moment we're only looking at the next game, but that's something that I've always done throughout my time here. The next game is the most important.
"Of course, we've got a big game at the weekend on top of that. But the idea is to go into the game tomorrow night to win and I'll pick the team that I think can win the game."
(Reporting by Sam Holden, Editing by Ed Osmond)
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