I am studying telecommunications and the textbook references the Royal Road and how couriers were expected to cross the entire 2700KM distance in 7 days when foot travel would normally take 90 days. In looking for more information regarding the Royal Road I aw Wikipedia states the couriers were held to slightly more reasonable 9 days.
All I can think of is ye olde pony express as they were able to do East to West in 10 days which is about 4400KM. Did they do it like this, just swapping horses and making mad dashes like a baton race?
So the most well known textual description comes from Herodotos, who provides his calculations pretty carefully. He gives the entire route in pasarangs, and explains that one pasarang equals thirty stadia. One stadion is about 190 meters, and Herodotos arrives at a grand total of 13500 stadia or 450 pasarangs. He takes 150 stadia (about 30 km) to be a day's journey, thus arriving at 90 days for the full 13500 stadia. He quotes an Aristagoras of Miletus speaking to a Spartan named Cleomenes that their journey to the king tooks three months. His account is detailed enough that he must have possessed some kind of document describing the route in detail.
Herodotos and Xenophon both do indeed describe a system of horse relays; Xenophon attributes the system to Cyrus, but it was probably only possible with the massive building projects undertaken by Dareios. Diodoros (not the most reliable source) also describes a system of voice relays, where messages could be shouted between guard posts. There are also attestations of beacon and torch signals.
I can't find any attestation of this express relay being expected to take any particular number of days, however. I can only imagine someone arrived at a 9-10 day number by dividing the 2565 km by the speed of a trot-canter at say, about 15 km/h giving about a 160 hour ride and then divided by the number of hours per day they could keep it up. However, Herodotos and Xenophon (who certainly knew his horsemanship) claims that they kept a horse per day of the ride whereas the pony express would switch horses every three hours or so. That means they could hardly cover more than 100 km or so per day. Xenophon adds however that they would sometimes switch to night-messengers. That would presumably about double the speed, so that could push it down to 12-15 days. If we use a shorter stadion of 160 m, then we get down to about 10 days.
So yeah, as far as I can tell there's no estimate for the shortest amount of time the distance could be covered in in the ancient sources. It seems like 25 days would be the approximate time for a typical royal messenger, and that that could be pushed down to about 13-15 days by riding at night as well.
I relied on Kuhrt's "A Corpus of Sources from the Achaemenid Period" for the ancient sources. If anyone is an expert on horsemanship, feel free to correct my estimates!