The negotiations, put a valuation of as much as $500 million on Twitter, which has become one of Silicon Valley's most closely watched start-ups, the paper said.
The talks, which were first reported by the AllThingsD blog, were confirmed to the paper by two people familiar with the situation.
Facebook offered to pay for the acquisition in stock, the paper said, citing a person close to the situation.
Putting a value on Twitter's shares proved controversial, according to the paper.
If it used the $15 billion valuation at which Microsoft Corp bought a stake in Facebook last year, it would have valued the Twitter purchase at $500 million, though that investment was seen as a high-water mark for Web 2.0, the paper said.
One person close to the situation suggested to the paper that the $15 billion valuation for Facebook was the top end of a range of values the two companies talked about, implying that a deal might have valued both Facebook and Twitter at a much lower level.
Biz Stone, Twitter's co-founder, would not comment on the talks, but suggested that the company wanted to remain independent to build on its messaging service, the paper said.