Hodgson steered Fulham to their first ever European final in May before they were beaten by Atletico Madrid and has emerged as strong favourites with some bookmakers to replace Rafael Benitez ahead of Aston Villa manager Martin O'Neill.
"He has done a remarkable job at the club during the past two and a half years.
"It doesn't surprise me if a number of clubs would be after him as a manager given his superb record."
Schwarzer, currently preparing in South Africa for the World Cup with his country, admitted: "The gaffer is at that stage of this career where he is thinking 'will this sort of opportunity come around again?'
"I suppose in a lot of ways you can't blame the guy if an opportunity like this came along.
"We hope he doesn't go, we hope he stays at Fulham, but we will have to wait and see what happens.
"But I wouldn't blame him at all (if he decides to move), not at all.
"It is Liverpool you are talking about, a huge club, who normally compete in the Champions League every season.
"There are four or five such clubs in the country and there would be very few managers or players who would find it difficult to resist going to them."