The 34-year-old is a free agent after leaving Crystal Palace last month and Villa, together with Chelsea and Burnley, have been credited with an interest.
Villa saw Tom Heaton depart at the end of last season and their need for a new keeper this summer will depend on the future of their current third-choice Lovre Kalinic.
The Croatia international, who has made just eight appearances since joining in a £7million move two-and-a-half years ago, spent the second half of last season on loan at boyhood club Hajduk Split.
Though the latter are thought to be keen on making the deal permanent, nothing is likely to be decided until Kalinic, part of Croatia’s Euro 2020 squad, returns to Bodymoor later this month.
Should he depart, Villa will look to recruit a keeper to provide competition for No.1 Martinez and Jed Steer.
Wales international Hennessey started his career at Wolves and made more than 150 league appearances before moving to Palace for £3m in 2014.