Simpson, who hit a first round 67, began his round with a duo of pars and a bogey on the third.
The 29-year-old North Carolina native hit a birdie on the fifth, but followed that up with his second bogey of the round.
However, that was the end of his bogey woes as Simpson hit a birdie on the seventh and began to regain his confidence when he sunk a 56-footer to record a birdie on the ninth.
Simpson enjoyed a lot more success on the back nine as he hit a birdie on the 10th, 14th, 15th and 17th, which came courtesy of an incredible 67-foot shot.
Streb was on song during his front nine as he hit three birdies on the fifth, seventh and eighth.
His solid performance continued into the back nine as he recorded his fourth birdie of the round on the 13th.
But, he ended on a low as he hit his sole bogey of the round on the par-four 16th.
Sitting two strokes behind Simpson and Streb are Martin Flores and Patrick Rodgers.
Flores hit eight birdies, including four in a row from the fourth to the seventh, and three bogeys during his second round 67.
Rodgers recorded five birdies and a bogey during his second successive round of 68.
World number one Rory McIlroy finished tied fifth and remains three strokes behind Simpson and Streb after his bogey-free round of 67.
McIlroy remained consistent throughout the day as he hit five birdies, one of which he managed to achieve by sinking a 29-footer on the 10th.
Meanwhile, defending champion J.B. Holmes was cut after faltering in his second round with a disappointing 76.
Holmes' struggles began when he hit a trio of bogeys from the second to the fourth.
Things went from bad to worse when he registered a double bogey on the par-five seventh.
Holmes' luck improved slightly when he hit two birdies in a row on the ninth and 10th.
But, his woes continued midway through the back nine as he recorded two more bogeys on the 13th and 14th.
Despite following the duo of bogeys with a birdie on the 15th, Holmes was unable to keep himself in contention and retain his Wells Fargo crown.