Archer will complete a memorable season when England under-21s face Slovenia in Huddersfield, having scored three goals in his last two outings for the Young Lions.
The 20-year-old has netted 23 times in all competitions including his first senior goals for Villa, while he scored seven during a loan spell with Preston.
The latter are keen to take him back to Deepdale next season, while Middlesbrough and Rangers are also keeping tabs on his situation.
But Villa boss Gerrard is keen to take a closer look at Archer during pre-season before deciding on his next move. The academy product signed a contract through to 2025 in December and is highly-rated at Villa Park.
Winger Trezeguet has, meanwhile, emerged as a target for Trabzonspor as he seeks a Villa exit.
The Egypt international made just one substitute appearance in the Premier League this season and spent the second half of the campaign on loan at Istanbul Basaksehir.
Trezeguet is one of several players who will be allowed to depart this summer as Gerrard looks to revamp his squad.
Villa have already made four new signings after completing deals for Philippe Coutinho, Boubacar Kamara, Diego Carlos and Robin Olsen but there is an increasing focus on moving players surplus to requirements off the books.