The Norwegian, who has come through the academy at Derby, is expected to go straight into the Brewers' squad for their game at Fleetwood on Saturday afternoon after being registered before the midday deadline.
Rams manager Nigel Clough said: "This is a good opportunity for Mats to go out and gain some experience of being in-and-around the first-team environment on a matchday.
"It's another step forwards in his progression and we're sure he'll benefit from it as a result."
Morch has yet to make his debut for Derby after signing a professional contract last season but he has gained match experience in Under-21 side during the past year.
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- Nigel Clough