The 30-year-old Canadian-born player, who appeared 42 times for England - the first 34 of those when Eriksson was manager - was released by Manchester United at the end of last season after a series of injuries which blighted his career.
Earlier this week he posted videos on YouTube on him carrying out exercise and mobility drills to prove his fitness and Eriksson was impressed enough by what he saw to give him the chance of a fresh start with the ambitious Championship club.
"I saw his videos and if he wants to come here I will open the door and stand and wait for him in the door," said Eriksson.
"He's been very unlucky in recent years but if he's close to what he was he is a fantastic football player. "
Hargreaves, who first came to prominence at Bayern Munich, played only five minutes of first team football for United in the last three seasons and is a free agent.
Leicester, who are attempting to get back to the Premier League for the first time since 2004, have had a busy summer after finishing 10th last season.
On Thursday they broke their club transfer record when Reading defender Matt Mills joined for 5.5 million pounds, while other signings include goalkeeper Kasper Schmeichel from Leeds United, defender Sean St Ledger from Preston North End and midfielder Neil Danns from Crystal Palace.
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