It is understood the new offer was a basic £35.5m, which would have risen to nearer £40m when various extras were added.
The Reds value the 20-year-old, who has also been linked with Arsenal and Real Madrid, at £50m.
Last week, Liverpool turned down a bid of £25m, plus add-ons, from Premier League rivals City.
Sterling joined Liverpool from QPR in 2010 and is under contract until 2017, but has rejected the offer of a new £100,000-a-week deal.
Liverpool manager Brendan Rodgers said last month he expected Sterling to stay at Anfield for the remainder of his current deal.
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Liverpool have formally rejected a second bid from Manchester City for England forward Raheem Sterling.