Singh, who won the PGA Championship last month, fired an 8-under 63 to take the lead midway through the Deutsche Bank Championship. He stands at 11- under-par 131 and leads by two over Woods, John Rollins and Bill Haas at the TPC of Boston.
If Woods does not finish in the top-seven, Singh will become No. 1. If Woods does finish within the top-seven, Singh would become No. 1 if he ties or beats him.
'I'm aware of it, but I'm not concerned about it,' said Singh. 'It's not going to affect me one way or the other if I overtake him or not. Yeah, if I win this tournament, I'll be No. 1; fine. But what would that change for me? I'll go out there next week and do it all over again.'
Woods posted a 3-under 68 on Saturday, while Rollins carded a 66 and Haas, son of Ryder Cupper Jay Haas, shot a 7-under 64.
Ryan Palmer, who shared the first-round lead with Woods, shot a 2-under 69 and is tied for fifth place with Shigeki Maruyama (66) at 8-under-par 134.
All players are chasing Singh as he chases Woods in the World Rankings.
Singh got off to an amazing start in Saturday's second round. He holed out from 91 yards for an eagle at the first, then ran home a 6-foot birdie putt at No. 2.
'With a start like that, I guess you can be very aggressive,' said Singh. 'I was feeling pretty good out there when I started off. '
His run of strong golf continued at the fourth when his pitching wedge approach stopped 4 feet from the hole. Singh converted the birdie try, then two-putted from 20 feet for birdie at the seventh.
Trouble loomed for Singh at the par-3 eighth. He three-putted from 30 feet for a bogey, but still made the turn at 4-under 32.
Singh regained his form at the 10th when his 9-iron second shot left him with a 15-foot birdie putt. He made that, then collected back-to-back birdies from the 12th, including a tap-in at 13.
He missed the green at the par-3 16th, but putted in for birdie from 30 feet. Singh went long with his approach at 17 and he putted down to 4 feet. He missed the par putt, but still had a one-shot lead with a chance to extend it at the par-5 closing hole.
Singh found the fairway at 18, then hit a 6-iron 15 feet over the flag. His eagle putt died right, but Singh tapped in for birdie and the two-shot advantage.
'I've been driving the ball well so far, if I keep doing that, you know, my chances are good,' said Singh. 'Ever since the move to the conventional putter, I have a lot more confidence in my putts. I'm putting well and that's giving me a lot of freedom to attack the pins and be aggressive on my chip shots.'
Woods reached the green in two at the par-5 second. He two-putted from 15 feet for the birdie, but found trouble two holes later. Woods drove into the hazard on the right and took a drop. He hit a 6-iron to 20 feet and two-putted for the bogey.
Woods soared up the leaderboard with his play late on the back nine. He got a favorable bounce from the right rough at 13 and made birdie from 9 feet. Woods failed to land in the fairway off the tee at 15, then missed the putting surface with a 9-iron. He chipped in from 15 feet for his third birdie of the round.
The unofficial tournament host missed the green at 16 and never recovered, bogeying the hole and falling to 7 under for the championship. Woods made an improbable birdie from close to 60 feet at the 17th and birdied 18 to get into a share of second place.
'I thought I played decent,' said Woods. 'I didn't play as well as I did yesterday, but controlled the ball where I missed the ball properly. Didn't make any putts on the front nine, but I got it going on the back.'
With Woods' run of 264 consecutive weeks atop the World Rankings in jeopardy, Woods has a simple philosophy.
'The No. 1 ranking, it takes care of itself just by winning tournaments,' said Woods, who has only one victory this season, the WGC-Accenture Match Play Championship. 'If you win consistently, you don't have to worry about it.'
Rollins, who won the Canadian Open two years ago, tallied three birdies on the front side, then mixed three birdies and a bogey in a five-hole span on the back to get into a share of second.
Bill Haas collected six birdies on the back nine and is in position for win No. 1 on the PGA Tour. Will he talk to his father, who is competing on the Champions Tour at Pebble Beach, for advice on what it takes to get into the winner's circle?
'He won't say much,' said Bill Haas. 'He's letting me learn it on my own. He learned it on his own. There's nothing that he can say to me that's going to make me play any better; it's all me.'
Charles Howell III (68) and Frank Lickliter (67) share seventh place at 7-under-par 135, while defending champion Adam Scott (67), Hank Kuehne (68) and Jay Williamson (68) are a stroke further back at minus-6.
The 36-hole cut fell at even-par 144 and David Duval made his first cut since the FBR Capital Open last June. He two-putted from 40 feet at the 18th to shoot a 1-under 70 and make the cut on the number.
'I was nervous a little bit,' said Duval, who was the last player to be ranked No. 1 before Woods in 1999. 'It's a different nervous. I shot 59, I've won an Open and played Ryder Cups. I was like, 'come on, you can two-putt.''