Autosport said the 27-year-old was the favourite to replace Gasly for the race at the Circuit of the Americas track, where Hartley won an endurance race last weekend.
Hartley is contracted to Porsche in the World Endurance Championship (WEC). A Porsche spokesperson told dailysportscar.com that "any request [to race F1] would be no problem".
It's also believed Hartley was close to a deal to race IndyCars alongside fellow New Zealander and four-time series champion Scott Dixon at Chip Ganassi Racing.
Toro Rosso have another standby option in former driver Sebastien Buemi, who has won the Formula E electric title since he last competed in Formula One in 2011.
Hartley has focused on endurance racing in recent years after his early bid for Formula One was thwarted as a teenager.
He was groomed by Red Bull from the age of 14 when he was plucked from Palmerston North.
As an 18-year-old, he tested for the Red Bull F1 team and was their reserve driver in 2009-10 before being dropped from the programme.
He went on to fashion a great record in GT racing, including his win with the Porsche factory team at Le Mans this year.
He is part of a trio including fellow Kiwi Earl Bamber and German Timo Bernhard that leads the WEC but will be looking for another drive next year after Porsche's decision to withdraw at the season's end.
Legendary Kiwi driver Kenny Smith, who mentored Hartley as a youngster, said the rising star had a childhood dream of racing in F1.
"He is one of those who is so competitive in everything he did, he wanted to be a winner and that is what has given him the drive to keep on going," said Smith.
"I could see that when he was 12 years old driving Formula Vees, driving off the track and doing all sorts of crazy things, he tried so hard -- and Brendon never gave in.
"He was always looking to go quicker and look what he has got now. It is a dream come true for a Kiwi to do what he is doing.
"It has been so good to be associated with him over the years."
