Cameron Young, the runner-up at St Andrews last year, leads the chasing pack, with Masters champion Jon Rahm, who carded a course-record 63 on Saturday, six shots back.
Unheralded American Harman, who was joint second at the 2017 US Open, has not lifted a trophy on the PGA Tour for six years.
But the world number 26, nicknamed the "Butcher of Hoylake" due to his love of hunting, has carved up the Royal Liverpool course during the first three days to pull clear of the field at 12 under par.
"You'd be foolish not to envision (it)," said Harman on the prospect of lifting the Claret Jug.
"I've thought about winning majors for my whole entire life. It's the whole reason I work as hard as I do and why I practise as much as I do and why I sacrifice as much as I do.
"If that's going to come to fruition for me, it has to be all about the golf. It has to be execution and just staying in the moment."
Not since French golfer Jean van de Velde's infamous collapse at the final hole of the 1999 British Open has a player squandered a five-shot lead going into the final round of a major.
Harman showed mental strength on Saturday, bouncing back from two early bogeys to card a two-under-par round.
But with rain lashing the Wirral coast, the conditions could yet provide a twist in the tale.
Rahm and Hovland find their groove
Young, also seeking his first major, shot a seven-under-par round on the final day of last year's British Open, losing out to Cameron Smith by a single stroke.
"With the lead he has right now, it's not necessarily going to be up to me," said Young. "It's just really time for me to focus on myself and see where that gets me."
Rahm had a mountain to climb after rounds of 74 and 70 on the opening two days, meaning the Spaniard only narrowly made the cut.
The world number three showed his class on Saturday with six birdies on the back nine in what he described as his best-ever round on a links course.
Rahm will be joined by world number five Victor Hovland, who starts at five under, in the penultimate group.
The Norwegian, who also narrowly missed out last year, finishing in a tie for fourth, found his groove in a third-round 66.
Home favourite Tommy Fleetwood is also seven shots back, but his chance may have gone in a level-par third round as others made their move.
Further back, world number 561 Alex Fitzpatrick is aiming to upstage older brother and 2022 US Open champion Matt as he starts at four under par.
The world number nine is at two under, but has instructed his parents to follow his younger sibling in his first major championship.
