United States midfielder Donovan, 29, was a huge hit with fans during a brief spell on Merseyside in 2009-10.
Moyes said: "It's something we will probably try and look to see if there's an interest.
"He was popular here, and we could do with some speed and width on the right.
"It wouldn't be something we'd be against, that's for sure. It's unrealistic to sign him on a permanent deal, though."