The venture-capital firm had committed to invest up to $20 million in Snapchat in May, the Journal cited one of the people as saying, without elaborating.
At least one other strategic investor has committed to joining the latest round of funding, the Journal cited two of the people as saying.
If the deal goes through, Snapchat would join a select club of tech startups with valuations of $10 billion or more, including car-ride service Uber and rooms-to-let startup AirBnB.
Snapchat, a so-called ephemeral messaging service popular with teenagers, turned down a $3 billion acquisition offer from Facebook Inc last year, a source close to the matter told Reuters.
It now lets users send photo-messages that vanish within seconds, but is expected to soon begin offering advertising or branch out into additional services. Snapchat is similar to a new crop of popular mobile messaging apps that compete with established Internet services such as Twitter Inc's and Facebook.
Though few of them have established business models, their rapid user growth and perceptions of advertising potential have aroused intense investor interest over the past year or so.
Neither Snapchat nor Kleiner were immediately available for comment.
© Thomson Reuters 2014