The 19-year-old has been training with the Rams and featured in two games during the club's pre-season tour of the south west last week.
Randall was hoping to secure a temporary move to Pride Park as he looks to gain further first-team experience but boss Nigel Clough will not be following up his initial interest.
"We have decided not to pursue anything with Mark. We have spoken to (head of youth development) Liam Brady at Arsenal and to Mark and although he did quite well for us while we were away, we think we need a different sort of midfielder," said Clough.
"We need somebody who is going to come in and really challenge the three central midfielders we already have in there.
"We have three very good players - Robbie Savage, Stephen Pearson and Paul Green - and arguably they have impressed as much as anybody in the last three or four weeks."