Citing unspecified documents and statements, the paper said Undercover Global Ltd, which was responsible for security at the embassy while Assange was staying there, sent the US intelligence service audio and video files of meetings he had with his lawyers.
The reports were allegedly handed over by David Morales, who owns the company and is currently being investigated by Spain's National Court, the paper said.
Speaking to AFP, one of Assange's lawyers confirmed the National Court was looking into the matter.
"There is a criminal case under investigation at the National Court but it is being conducted in secret... and we cannot say anything about what is being investigated beyond what has been leaked" to the press, Aitor Martinez told AFP.
The leak "probably came from employees at the firm", he said.