Williams, 19, has started five of United's last six matches, and was again impressive in Wednesday night's 1-0 win over Wolves in United's FA Cup third round replay.
In Williams' last seven matches, United have kept six clean sheets - the 3-1 loss to Manchester City being the exception.
Scholes believes that, allied to Shaw's recurring injury problems, the youngster is making a first-team spot his own.
“You'd have to choose him [Williams] at the moment," said Scholes. "I do like Luke Shaw but he just can't string a run of games.
“He can't get that fitness which he needs to play for Manchester United and be up and down the wing all the time.
“At the minute Brandon Williams is doing that so he is in pole position. He's an excellent defender but he can also get forward as well.
“He's got quality, he's a talented lad and he'll get up and down the pitch. He's got two good feet, I think he's more of a right-footed left-back which is strange.
“He's got the quality to get forward and the quality to score goals."