The Germany U-21 international midfielder only joined the Reds over the summer in a deal worth around £10m from Bayer Leverkusen, but his playing time has so far been limited.
A goal against Chelsea has been the highlight of his brief Merseyside career so far, with his last outing, as a substitute, against Burnley spent at centre-back.
Despite this, Brendan Rodgers is known to be a big admirer of the youngster, but Moyes is hopeful of securing a temporary deal for him.
The ex-United and Everton chief is now in charge of Spanish side Sociedad, and can offer Can regular action.
Liverpool’s rumoured move for Lazio midfielder Ogenyi Onazi may also be a boost for the Scot, with the Nigerian likely to command first-team action.
Moyes also wants to take United youngster Adnan Januzaj on loan for the remainder of the campaign.