Pearson has made a big impression at Gresty Road since his arrival last month.
The striker scored the winning goal on his debut at Oxford and found the back of the net again in a 4-1 victory over Dagenham. His other appearance at Barnet was cut short as he was sacrificed after Adam Dugdale and David Artell were sent off.
Pearson's deal expires on February 20 but Crewe Steve Davis told his club's official website: "We would like to keep Greg and I think he is keen to stay here as well.
"He has enjoyed it and hopefully we can extend his loan. Of course, he belongs to Burton and we will have to speak to them to see what they want to do.
"He can't play against them next week and I am sure I will have the chance to speak to (Burton boss) Paul Peschisolido then."
