The Gunners have a specialist at their helm in the form of Unai Emery, while the Blues are seeking major silverware in Maurizio Sarri's debut campaign.
Arsenal have been drawn against French side Rennes in the last 16 of the Europa League.
Chelsea will face Dynamo Kiev of Ukraine as a result of Friday's draw in Nyon, Switzerland.
The first legs of the last-16 ties are set for March 7, with the returns taking place on March 14 as both Premier League clubs look to stay on course for the final in Baku.
Arsenal progressed from the round of 32 with a 3-1 aggregate win over BATE Borisov, overturning a first-leg loss in Belarus with a 3-0 victory at the Emirates Stadium on Thursday night.
Chelsea, meanwhile, defeated Malmo, winning the second leg 3-0 at Stamford Bridge to add to a 2-1 away triumph in Sweden.
Both English clubs will be at home in the first leg.
Friday's draw was not protected by any seeding, but teams from Russia and Ukraine were kept apart. Dynamo Kiev were first out and drew Chelsea.
Elsewhere, Eintracht Frankfurt of Germany were paired with Serie A side Inter Milan, Dinamo Zagreb will play Portuguese outfit Benfica, while Napoli tackle Austrian side Red Bull Salzburg.
Valencia, who knocked out Celtic, have been pitted against Russian club Krasnodar, while former winners Sevilla face Slavia Prague and Zenit St Petersburg play Villarreal.