The Spanish publication says that the Blues are in the "pole position" to acquire the Spaniard as Antonio Conte's replacement but the Gunners could join the race if they decide to part ways with Arsene Wenger at the end of the current season.
But Mundo Deportivo says that the Frenchman's future at the Emirates Stadium will be reviewed at the end of the current season.
And if Arsenal fail to achieve their targets the Spanish publication understands that Chelsea target Luis Enrique will be one of the main candidates to take over Wenger.
Arsenal recently appointed former Barcelona chief Raul Sanllehi as the new head of football relations and Mundo Deportivo believes that they could use that to join the battle for Enrique.
The former Celta Vigo and AS Roma boss worked with Sanllehi at the Nou Camp and could be tempted to reunite with him in the Premier League if there is a vacant position at the Emirates Stadium, the Spanish report suggests.
Mundo Deportivo says that Paris Saint-Germain have also tied to tempt Enrique to the club but the former Barcelona boss would rather move to the Premier League than Ligue 1.
The report predicts that there could be a summer battle between Chelsea and Arsenal to appoint Enrique with the Blues currently leading the race.
Earlier this week, Mundo Deportivo rivals Sport said that the Blues had already opened talks with the Spaniard to replace Conte before the end of the current season.
Sport said that the Italian will continue at Chelsea for now but added that the club are likely to sack him if the west Londoners suffer defeat against West Bromwich Albion in the Premier League on Monday (12 February) or Hull City in the FA Cup fifth round.
However, Mundo Deportivo says that Chelsea owner Roman Abramovich is determined to keep Conte until the end of the season.
The reports adds that they are already preparing for life without Conte and Luis Enrique is a serious candidate to replace him ahead of the 2018-2019 campaign.
Meanwhile, they added that both Chelsea and Arsenal also admire Massimiliano Allegri with the current Juventus manager being also among the potential candidates to replace Zinedine Zidane at Real Madrid if the Frenchman leaves the Santiago Bernabeu at the end of the season.