The former striker is well known to English audiences after spending the 1994-95 season in the Premier League with Tottenham.
As a coach, Klinsmann, 51, took Germany to the semi-finals of the 2006 World Cup, held on home soil.
An ill-fated stint in charge of Bayern Munich ended in April 2009 when he was sacked and replaced by Jupp Heynckes.
He took over the US side in 2011 and won the Concacaf Gold Cup in 2013 and reached the last 16 of the 2014 World Cup, where they were beaten by Belgium in extra time.
USA were beaten by Jamaica in the semi-finals of last year’s Gold Cup and reached the semi-finals of last month’s Copa America Centenario, where they lost to Argentina.
Selecting a new England boss
Hodgson’s successor will be chosen by a three-man panel of FA technical director Dan Ashworth, chief executive Martin Glenn and vice-chairman David Gill.
They have already spoken to former internationals – including Frank Lampard, Rio Ferdinand and Gary Lineker – as part of its review and have not ruled out an interim appointment, with World Cup 2018 qualifying beginning in September.