Durham leg-spinner Borthwick was scheduled to arrive in Melbourne late Monday and will train with the squad before the fourth Test starts on Thursday at the Melbourne Cricket Ground.
Team officials said Borthwick, recommended by Swann for the England team, would be available for selection in the Boxing Day Test.
Swann announced his immediate retirement from international and first-class cricket mid-way through the series, with Australia already holding an unassailable 3-0 lead and retrieving the Ashes urn they lost in 2009.
"Personally I hope Scott Borthwick gets a chance before long," the off-spinner said at his Sunday press conference to announce his retirement.
"He's a leg-spinner with a bit of X-factor and he can bat as well."
Kent off-spinner Tredwell will arrive in Melbourne during the fourth Test and will be available for selection for the final Test in Sydney beginning on January 3.
Monty Panesar, who is expected to be part of the England attack in the Melbourne Test, paid tribute on Monday to fellow spinner Swann.
He noted Swann took 255 Test wickets compared to Australian legend Richie Benaud, who snared 248 with his leg-spin.
"That's the elite class ... he's obviously an icon cricketer, (he's) been a match winner for England and probably one of the best spin bowlers in the modern game," he told reporters.
"We loved him to bits when he played with us -- he had great character and his sense of humour was good."
"We are all disappointed, the way the results have been."
Panesar said the English take great pride in representing their country.
"When you're playing for England, it is the pinnacle for every sportsman," he said.
"In no way (do we) take it for granted, absolutely not."