The Daily Star says Sheihk Mansour has sanctioned a move for Lampard.
The super-rich Sheihk bought an 80 per cent stake in the A League club for Â£6.7m in January.
Lampard is out of contract at the end of the season and is already being courted by LA Galaxy.
Melbourne are offering Â£60,000 a week after tax for two years, with the option of a third year.
The 35-year-old, though, is hoping to secure another year's extension with the Blues and is unlikely to take up the Aussie option.