According to James Nursey of the Daily Mirror, the Red Devils have not approached Shaw over a contract extension, with his current terms set to expire in 2019.
"Shaw is keen to secure his long-term future at United, but it looks like he may have to wait for a new deal that could see his £80,000-a-week wages brought into line with United's other senior stars on around £120,000 to £150,000," it's noted.
Nursey added the Gunners and Blues have both had their interest piqued as a result, as both clubs are said to be considering moving for a left-back at the end of the season. United, meanwhile, will reportedly move for Tottenham Hotspur's Danny Rose.
As noted in the report, it appeared as though Shaw had no future at United earlier in the campaign, as he was marginalised by manager Jose Mourinho. However, the England international has forced his way back into contention and has been a regular for the Red Devils in recent weeks.
The Portuguese coach has been critical of Shaw in public, casting further doubts over his long-term future at United.
But as relayed by the Red Devils' Twitter account, Mourinho has been full of the praise for the manner in which Shaw has kicked on:
Given how the 22-year-old appears to have finally recovered from his injury woes and added some consistency to his game, it would be a big shame for many associated with United to see him go.
After all, he provides a crucial balance to the United XI when he is in the side. Shaw is always seeking to get forward, boasts tremendous energy and his delivery is often on-point when he does get into the final third. It feels as though genuine progress has been made by the left-back.
With that in mind, it's a surprise to read United haven't moved to offer him a new deal quite yet, especially given how little there is to run on his current one.
Arsenal and Chelsea would potentially benefit from his arrival, too. As noted by Nursey, the Gunners may be on the hunt for a successor for Nacho Monreal, who is 32, while it's also said there is some speculation linking Chelsea's Marcos Alonso with a move back to Spain.
Should United go for Rose, Shaw would surely be bumped down the pecking order again, and that would be a concern for a young footballer who needs regular minutes at this stage in his career.