Baaeed, officially rated the world's best horse, won the Juddmonte International in imperious fashion under Jim Crowley at York, beating Mishriff by six and a half lengths on what was his first start beyond a mile.
Sheikha Hissa, who now heads the Shadwell operation established by her father Sheikh Hamdan Al Maktoum, has decided the son of Sea The Stars should have just one final run before retirement.
Connections had previously stated their intention to go to Ascot for the Qipco Champion Stakes on October 15 for Baaeed's career swansong.
However, after his scintillating display over 10 furlongs on the Knavesmire, the possibility of a further step up to 12 furlongs in the Arc on October 2 - for which he must be supplemented - has not been ruled out.
And six-time champion jockey Fallon is adamant Baaeed has the ability to end his career unbeaten.
"Of course he can win an Arc," said Fallon. "I think he has the pedigree to stay a mile and a half, although I haven't gone into it, but I know he has the pedigree to stay a mile and a quarter.
"He travels for fun and wasn't stopping at York."
Fallon partnered Hurricane Run to victory in the 2005 Arc for trainer Andre Fabre and took the prestigious Group One prize again two years later with with Dylan Thomas for Aidan O'Brien.
Though he knows there are obvious pitfalls, not least the possibility of autumn ground and a bad draw, the 57-year-old feels there is another factor Baaeed must overcome, if Paris is the chosen destination for him to end his career with an 11th straight win.
He added: "I think the ground might be a slight concern, as he wouldn't want soft ground, would he?
"But the thing you have to look at is this - he hasn't been trained with the Arc in mind.
"I have been with a couple of good trainers and have won the Arc a couple of times, but you need to be training the horse for a long-term target. Aidan trains his horses for such races and gives them mid-summer breaks.
"I don't think the Arc was a target - the Champion Stakes was always going to be his target.
"So that is a box you couldn't really tick.
"But he has got all the rest of it. Ability-wise, he has everything. The ground and the trip are questions and the draw might be an issue. But he has all the talent in the world.
"The Arc comes two weeks before the Champion Stakes, though. At that time of the year, horses who have been on the go all year can sometimes go over the top.
"So it wouldn't be a shoo-in, but if he had been trained for the race, I could see him winning, of course I could.
"Of course you couldn't oppose him if he ran in an Arc, you know? He is the one I'd want to be on - although the first thing I'd be asking in any Arc is, 'what has Andre Fabre got in it?'.
"I don't think Aidan has anything really this year. Pyledriver is like the housewives' favourite, but I don't think he'd beat Baaeed."
Fallon, now a key contributor to Charlie Appleby's success at Godolphin as a work rider, is also intrigued by what Baaeed has meant to the trainers' championship standings, with Haggas and Appleby neck-and-neck at the top.
"I know William would love to have that on his CV," said Fallon. "He is a lovely guy. They both are. I was riding out for William around the time I retired.
"I thought then, he could be a champion trainer. He had some lovely horses and is a good trainer as well. He puts everything into it, him and Maureen. It is their life.
"I do think he will be champion trainer one day. William has good owners and good horses and he deserves it if he wins it.
"He is up there and if he wins the Champion Stakes, that would probably put him a good bit in front.
"He probably doesn't have the same quality of two-year-olds that we have - Charlie has got serious two-year-olds, but he has Baaeed and I definitely think he can retire unbeaten."