Borsheim had five birdies and Rumford six birdies and one bogey as they shot 5-under-par 67 at the Gleneagles Hotel in Scotland.
Five players, including Paul Lawrie, are tied for third at 4-under. None of those players was bogey-free Thursday but Lawrie had to overcome three bogeys to stay within one stroke of the leaders.
Five more players are at 3-under.
Borsheim had a very steady round. He had a birdie at the second hole and made the turn at 2-under after adding a birdie at No. 7. He had a birdie at 10 and got into a share of the lead with a birdie at 14. Borsheim matched Rumford with a birdie at 18 for the first-round lead.
Rumford had back-to-back birdies at Nos. 2 and 3 and again at 8 and 9. He made birdie at 16 before his lone bogey at the 17th hole. He also closed with a birdie.
They were among 55 players who had birdies at the 513-yard, par-5 18th.