The former West Ham United captain was absent from the 1-0 loss to Inter Milan in the Champions League on Wednesday due to the problem, which he picked up in the identical defeat to Newcastle United last weekend.
Gunners head coach Mikel Arteta was coy on the severity of Rice's problem when asked ahead of the trip to Milan, although it was understood that he was not suffering from anything too severe.
As a result, the ex-Chelsea academy star was said to have been in with a good chance of making an immediate comeback in time for the showdown with Enzo Maresca's team this weekend, but according to the Daily Mail, he is suffering from a broken toe.
Nevertheless, the report adds that Rice will be given every possible chance to work his way back to full fitness for the London derby, and Arteta will wait until the last minute for him to make himself available.
Rice to play through the pain barrier against Chelsea?
Arsenal come into Sunday's crunch clash seven points below leaders Liverpool in the Premier League table, having failed to win any of their last three contests in the top flight, taking just one point during that miserable run.
The Gunners' struggles have allowed Chelsea to jump ahead of Arteta's team in the rankings, although only goal difference is keeping Arsenal outside of the top four, and the report suggests that Rice would be prepared to play through the pain barrier.
Earlier on Thursday, Rice was included in Lee Carsley's final England squad for their Nations League clashes with the Republic of Ireland and Greece later this month, suggesting that his problem is not overly severe.
The 25-year-old has failed to hit the same heights he managed during his debut campaign in North London, but he has been a vital player for the Gunners nonetheless, scoring one goal and setting up two more in 13 games in all tournaments.
However, Rice's only direct contribution in the Premier League season so far was an assist for Mikel Merino in the 2-2 draw with Liverpool last month, and he spent seven years in the Chelsea system from 2006 to 2013.
What other injury concerns do Arsenal have before Chelsea?
Rice is one of a couple of doubts for Arteta ahead of Sunday's derby, as another erstwhile Blue - Kai Havertz - sustained a nasty cut to the head in the loss to Inter in midweek after clashing with Yann Bisseck.
However, Arteta is optimistic that the Germany international - who was left bloodied by the incident - was not concussed and should therefore be fine to feature at his former stomping ground.
Merino is also expected to be OK, despite taking a whack to the head from Yann Sommer in midweek and coming off at half time, but Riccardo Calafiori (knee), Takehiro Tomiyasu (knee) and Kieran Tierney (hamstring) remain unavailable.
Arteta is also facing a huge dilemma over whether to start captain Martin Odegaard, who made his comeback from a severe ankle injury in the final few moments of Wednesday's European loss.