Gloom has enveloped Liverpool after an embarrassing League Cup defeat by League Two Northampton Town on Wednesday and last weekend's defeat by Manchester United. Saturday's result left the 18-times league champions languishing in 15th place.
Some 10,000 fans stayed behind at Anfield to protest against American co-owners George Gillett and Tom Hicks, and manager Hodgson said he sympathised with the supporters.
"I don't think anybody at the club wants anything but a solution to the ownership problem," Hodgson, whose side have managed just one win in six Premier League matches, said on the club's website.
"The fans here are very passionate and care an awful lot about this football club.
"You cannot criticise them for showing their displeasure about the situation, because like ourselves they want to see the club moving forward and not sort of stuck with owners who are trying to sell the club."
Gillett and Hicks put the debt-laden club up for sale in April but a prospective takeover by Chinese businessman Kenny Huang failed to materialise last month.
With uncertainty hanging over the club off the pitch and a playing squad that looks ill-equipped to break back into the top four of the Premier League, Hodgson faces a tough task and he called for patience on Saturday.
"It's quite simply the 'Rome wasn't built in a day' adage," Hodgson said after another scrappy performance. "There's lots of things I see in the game that I think we need to work on.
"We need to work more together, we have a lot of new players coming into team. We haven't had that much time with matches in the Europa League and the league.
"But I think that the more we work and the more the team plays together, the better we will become.
"These are games we all need to win, but the league is very, very tight. If we had won today we could have found ourselves in fourth or fifth place.
"I don't think we are playing like a team in fourth or fifth just yet, but that's how close the league is. There's not a lot between teams and it should be seen as how hard the Premier League is."
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