The 51-year-old Australian shot a four-under 68 on Saturday to get to nine under and is tied fourth heading into the final round of the Boeing Classic in Snoqualmie, Washington.
Allan has already won the Galleri Classic in March and Dick's Open in July.
However he will have his work cut out to catch 2009 British Open champion Cink, an eight-time PGA Tour winner, who posted a bogey-free 64 take a three-shot lead from New Zealand's Steve Alker (68) and fellow American Paul Goydos (69).
Cink moved up 10 spots from the first round on Friday with eight birdies at Nos. 6, 7, 8, 12, 14, 15, 16 and 18 to get to 12 under at The Club at Snoqualmie Ridge.
"If you stopped me after about No. 5 and said, 'Hey, what are the chances you're going to shoot 64 today?' I would have probably said you were crazy because nothing was happening at all," Cink, 52, said after the round.
"I made a really good par save on the third hole, which is a tough par 4. I made a sloppy par on the first hole, which is an easy par 5. ... Nothing was happening.
"But then, amazing how your momentum can flip with just one, really one putt, and a good putt on 6 from about 15 feet went in the centre. It was kind of just the lid was off."
Alker was working an unblemished round with birdies at Nos. 5, 6, 12, 14 and 17, then he bogeyed the par-5 No. 18.
"Yeah, just unfortunate about 18," Alker said. "Going to make dinner taste a little bit not as good. But I'm right there, so I like the golf course, the game's feeling OK, so go get 'em tomorrow."
First-round co-leaders Jerry Kelly and Matt Gogel each fired a 71 to move to 8 under and are tied for fourth with Allan.
Kelly, 58, is trying to win the Boeing Classic for the second time. It was the site of his first PGA Tour Champions victory in 2017; now he's searching for his 14th title overall.
Greg Chalmers was the next-best Australian, tied 19th at four under after a 71, with Richard Green (710, Brendan JOnes 73) and Stuart Appleby (73) two shots further back.
- with AAP
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