The message, sent from an e-mail account called "g20london riots", stated that it was aware of Sir Fred's current location abroad but made no further threats.
The name of the sender, however, adds further fuel to the theory that anti-capitalist protesters may be behind the attack.
The message was sent to the same Evening News address and used the same e-mail provider as the two which were received immediately following the vandalism in the early hours of Wednesday morning.
It comes as police continue high-visibility patrols outside Sir Fred's multi-million-pound home in The Grange area.
Officers said the area would receive "regular" visits, although a police guard has been withdrawn from the property.
With huge protests planned for London in the coming week to coincide with the G20 summit, the latest e-mail is now being studied by Lothian and Borders Police detectives, who are sharing intelligence with forces across the UK.
But they said it was "too early" to determine whether the e-mail was linked to the first messages from a group called "bankbosses arecriminals", which claimed responsibility for the attack.
Three ground-floor windows of the stone villa were smashed, while the rear window of a Mercedes S600 was broken, as well as a rear passenger window.
Detective Constable Simon Berwick, based at Craigmillar police station, said officers were working to identify the IP address of the latest e-mail.
He said: "The e-mail did not have immediately obvious details which we can progress, but we're taking a look at it now to see if we can find the IP address to determine where it was sent from.
"It's too early to say whether it is linked to the earlier e-mails."
A contact number given on the message turned out to be for a taxi firm in south-west London called Forest Hill Radio Cars.
Simon Martin, head of Forest Hill Radio Cars, said he could offer no explanation why the number was used.
A police spokesman said: "Regular patrols in the area will continue to provide a high-visibility presence for the time being."
He added that CCTV footage from Mr Goodwin's home and the surrounding area was still being examined by police.
Officers are offering security advice to other bank bosses living in the Lothians who fear they may be the next targets. And Edinburgh's Chamber of Commerce has issued a security alert to companies and is offering advice and helping to arrange for police visits where necessary.
RBS and other big Edinburgh-based financial giants are understood to be beefing up security at offices and homes of their top executives amid reports of death threats and abusive phone calls.
RBS paid 290 a month for security at Mr Goodwin's home, the bank admitted earlier this month. At the time the bank refused to comment on the details of the arrangement, saying that could compromise his safety.