Masters champion Rahm was in danger of missing the cut at one stage on Friday, but shot up the leaderboard by shooting the lowest round of the tournament with an eight-under-par 63 to get to six under.
The Spaniard moves into second place, one shot ahead of home favourite Tommy Fleetwood before he heads out with Harman in the final group at 14:30 BST.
Rahm was left frustrated by the brutality of Royal Liverpool's 82 bunkers during his first round and four short missed puts in his second as he sneaked into the weekend at two over par.
But the world number three showed his class on a day when low scoring has been aided by heavy overnight rain softening up the course.
Rahm picked up six shots in his final nine holes to ramp up the pressure on the unheralded Harman.
The American pulled clear of the field with a stunning six-under-par second round of 65 to move to 10 under.
But the 36-year-old's temperament is set to be tested over the weekend as he aims to claim a first-ever major championship and end a six-year wait to win any event.
"I've had a hot putter the last couple of days so try to ride it through the weekend," said the world number 26.
Harman will also have to cope with raucous support behind Fleetwood, who hails from Southport, just 30 miles (48 kilometres) from Hoylake, and is aiming to become the first Englishman to lift the Claret Jug since Nick Faldo in 1992.
"If somebody said you're going out in the last group on Saturday, I don't care what the situation was or what anybody had shot, I'd have probably taken it," said Fleetwood, 32.
"That's the way I look at it, and just look forward to playing over the weekend."
World number two Rory McIlroy picked up three shots in his opening five holes to close to within six of Harman and move into a share of fourth.
The Northern Irishman is aiming to end a nine-year major drought and repeat his sole British Open triumph the last time the tournament was held at Hoylake, in 2014.
World number one Scottie Scheffler will not be adding to his sole major at the 2022 Masters.
The American avoided missing the cut thanks to a birdie at the last on Friday, but a one-over-par round on Saturday pushed him to four over for the tournament.
PGA Championship winner Brooks Koepka is also at four over after a 72.
