Shaw extended his deal by a year in December but the club have moved to tie the left-back down to a longer contract. His deal was set to expire in 2024 but is now set to run until 2027.
The England international is approaching 250 appearances for United since arriving from Southampton in 2014. After being dropped earlier this season, he has established himself as Erik ten Hag's first-choice left-back at Old Trafford and was part of the team which lifted the Carabao Cup in February.
Ten Hag was asked about Shaw's contract at his weekly news conference on Friday, ahead of Sunday's trip to Newcastle.
"If he signed, I don't know, I haven't heard," the United manager said.
"I have to wait for that. If true, I am pleased. We definitely want to keep him because he's a really important player for our team."
Shaw's is the first of a number of contract renewals United hope to sort out before the end of the year.
Sources have told ESPN that talks are ongoing with Diogo Dalot and David De Gea. Dalot has a contract until 2024 while De Gea's contract will expire in June. United are also planning to step up negotiations with Marcus Rashford before the end of the campaign.
Meanwhile, United are hopeful Rashford will be available to face Newcastle after the striker pulled out of the England squad because of a side injury.
"I've good hope," Ten Hag said. "He trained but he had to do some treatment, he had to do some individual training, today [Friday] he joined in with the group. We have one more day. I have a good hope he will be available."