The duo are likely to continue training at London Colney alongside first-team option Folarin Balogun who spent the least season on loan at Middlesbrough.
Partey was left out of Arsenal's squad that beat Nurnberg 5-3 in Germany on Friday and will again miss out according to The Times.
The 29-year-old Ghana midfielder played 45 minutes in a 5-1 friendly win over Ipswich on July 2, returning for the first time since a thigh injury he picked up in April.
Leno will also remain in London with the goalkeeper close to a £10million move to Fulham after being replaced as the Gunners' back-up by US international Matt Turner.
The German signed from Bayer Leverkusen for £22.5million in 2018, but lost his place as Arsenal's no.1 to Aaron Ramsdale last season and is now to return to first-choice football.
Reports in Leno's homeland claim Arsenal are holding out for £11m for the 30-year-old, who is keen on a move to Craven Cottage after being offered a three-year contract.
Meanwhile Balogun also looks set to depart once again, likely on loan, with the 21-year-old making an impression during his temporary stint at Boro last season.
After Gabriel Jesus' hat-trick against Nurnberg, Arteta hopes to continue his side's progress as they head to the US to take on Orlando City and Chelsea in the Florida Cup.
They will then head back to the Emirates for their final pre-season game against Sevilla on July 30, before kicking off their Premier League campaign against Crystal Palace on August 5.