Northern Ireland's Rory McIlroy reacts during the second round
Jordan struck a stunning eight-under par round of 64 at the Old Course in St Andrews to move to 14-under, one shot clear of Scotland's Calum Hill, Swede Joakim Lagergren and fellow Englishman Matthew Southgate.
Rose is among a six-strong group two shot off the lead after a superb front nine at Kingsbarns.
The world number four carded an eagle and six birdies to make the turn in 28. A double bogey at the 11th halted his progress, but Rose made three more birdies in the final seven holes to position himself nicely with Saturday's round at St. Andrews to come.
McIlroy, who is playing alongside his dad in the pro-am tournament, also made a scintillating start with an eagle and three birdies in his first five holes at Kingsbarns.
One more birdie at the 12th and par golf the rest of the way saw the Northern Irishman complete a 66 to move to eight-under.
Two-time Alfred Dunhill champion Tyrrell Hatton also moved into contention with a four-under round at Kingsbarns to move to 10-under, while Open Championship runner-up Tommy Fleetwood is nine-under after a three-under round on Friday.
Each player plays a single round at St Andrews, Kingsbarns and Carnoustie on the first three days before the cut with Sunday's final round at St Andrews.