Shaw was withdrawn after injuring his ankle, while substitute Williams suffered gash to his head following an aerial collision with Kyle Walker-Peters.
"Luke twisted his ankle," said Solskjaer at his post-match press conference. "Hopefully, we can have some good news on him after we've done a scan.
On Williams, he added: '"I can't say [really], I've not seen Brandon because he was with the medical staff and I think he was getting some stitches with a gash over his eye.
"So let's see how he is."