Ponting sent out his appeal to Murali on his return home from the ill-fated Zimbabwe tour on Monday.
Reports from Colombo over the weekend said the Sri Lankan spin wizard had not changed his mind over quitting the upcoming tour of Australia, but he was under pressure to do so.
Sri Lankan cricket officials said Test cricket's greatest wicket-taker had not officially told the authorities about changing his mind after Australian Prime Minister John Howard upset him by calling him a "chucker".
But Ponting made a plea to Murali to put aside his ill-feeling and come and play against the Test champions Australia.
"I'd love to see him here. He's a fantastic player and opponent, and he's a great competitor," Ponting said at Sydney Airport on Monday.
Murali recently broke retired West Indian Courtney Walsh's world record of 519 wickets, with Australia's Shane Warne in close pursuit.
But Murali has been under scrutiny since English match referee Chris Broad reported his suspect action on the "doosra", a leg-break delivered with an off-break action, to the International Cricket Council (ICC) in March.
Ponting believes the Australian public would welcome Murali.
His presence would certainly add spice to the two-Test winter series with Murali (527 wickets) and Warne (517 wickets) going head to head.
"I'm sure the Australian public would love to see him here," Ponting said.
"Let's hope he tours. It would be a great challenge for us if he does because he's a very, very good bowler."