"The game in Melbourne was fantastic, and it was played at 100 kilometres an hour," he told the Herald on Sunday. "But the All Blacks are the best rugby team in the world and it would be great to see them do their stuff.
"I'd love to come down to New Zealand do a few songs before one of the big All Black matches but no one has asked me yet."
While on the flying visit to Melbourne, Sting took the opportunity to tell his army of Kiwi fans about his new album 57th and 9th.
Released next week, it is his first full-blown rock album in more than a decade. The title is a reference to the New York intersection he crossed every day to get to the studio where much of the album was recorded.
His new single I Can't Stop Thinking About You is reminiscent of The Police in their 1970s and 80s heyday.
Another song, 50,000, addresses the subject of the mortality of rock icons, and was partly inspired by the death of David Bowie, Prince, Motorhead frontman Lemmy and his good friend actor Alan Rickman.
Sting turned 65 while he was in Melbourne and was in a reflective mood.
"People think rock stars and cultural icons are immortal but the reality is we are all getting on a bit now," he said. "Turning 65 didn't bother me but I don't think I'd have the nerve to pick up my bus pass or state pension."
Sting last played in New Zealand in January on a double bill with Paul Simon at Auckland's Vector Arena.
He said he will gauge the reaction to 57th and 9th before deciding if he will support the album with a world tour, including gigs in New Zealand.
"I first came down here in the 1980s and I was fascinated by all the old British cars on the road, which I hadn't seen in many years," he said. "I always look forward to seeing that, even though it ages me a little bit more every time."
Win with the Herald on Sunday!
The Herald on Sunday has 10 copies of Sting's new album, 57th and 9th, to giveaway.
To be in to win, send an email to news@hos.co.nz, with the keyword 'Sting' in the subject field, and also include your name, address and phone number. Entries close at 5pm, Tuesday, November 8.