And last season he steered the Cottagers into the Europa League final - an achievement that led to him being named manager of the year.
Unsurprisingly, he has caught the eye of some top clubs and it appears Liverpool are keen on the former Inter Milan and Blackburn boss.
Schwarzer, who will be playing in the World Cup finals for Australia, added: "It doesn't surprise me if a number of clubs would be after him as a manager given his superb record.
"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.
"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."
And former Liverpool boss Gerard Houllier believes Hodgson would be a good choice for Liverpool.
He told BBC Radio 5 live's sportsweek programme: "Roy has done very well with an average team at Fulham.
"I would be delighted if he could coach Liverpool, because he deserves it.
"He is used to coaching stars, as well, because he was coach at Inter Milan."
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