The four-year-old extended his unbeaten record to 10 with an impressive triumph in Wednesday's Juddmonte International at York, switching up to a mile and a quarter for the first time and dishing out a six-and-a-half-length beating to defending champion Mishriff.
Bookmakers immediately started offering short odds about Baaeed for the 12-furlong Prix de l'Arc de Triomphe in October, but both Haggas and Sheikha Hissa were cool on the proposal immediately after the race.
Haggas has long had his sights set on the Qipco Champion Stakes as Baaeed's career finale, but his entry in the Royal Bahrain-backed Irish equivalent gives connections a further opportunity should they decide the son of Sea The Stars wants another run.
Haggas told Nick Luck's Daily Podcast: "He seems fine, he's trotted up good - he's lost a bit of weight but we're happy with him and very proud.
"Ultimately it is not my decision (where Baaeed runs). The horse doesn't belong to me, the horse is trained by us and it's Sheikha Hissa's decision and I'm sure we'll talk about it again because lots of people will be trying to convince Angus (Gold, racing manager) and Sheikha Hissa that the Arc is the right way forward.
"I'm open to suggestions, that's always been our plan to go to Ascot but plans can change. I'm not saying for a minute they're going to, but the only thing I will say is it's nearly two months now to Champions Day and that's quite a long time."
Of the 10-furlong Irish Champion Stakes at Leopardstown on September 10, Haggas added: "That's the perfect fit if he was to have another race, so that's obviously open for discussion.
"He's in the race and we'll just see what happens but he'd have to be in pretty rude health to be going to Ireland, so we shall see."
On the prospect of the Arc, Gold told Sky Sports Racing: "It was always William's plan to end up at Ascot. Obviously there was a lot of talk about the Arc yesterday and would we consider that.
"I spoke to Sheikha Hissa briefly after the race and asked if she would even be thinking of the Arc, but she said she thought it best to stick to William's plan and end up at Ascot so for the moment we'll stick with that.
"I'm sure she will talk to her family when she gets home and there'll be another discussion, but that is what she said yesterday and unless William changes his mind I'm sure that is what she'll want to do."