The Edinburgh club are interested in the 25-year-old but face competition from a number of teams both north and south of the Border.
Findlay’s Kilmarnock contract expires at the end of May and Hearts believe the left-sided central defender would be a useful addition to their back line.
He is entitled to talk to any interested club about a pre-contract as he is available for free this summer.
Hearts hope he can be persuaded to continue his career at Tynecastle Park next season, by which time they hope to be back in the Premiership.
Other Scottish and English clubs have also expressed an interest in Findlay, meaning he is likely to have a number of options when the time comes to decide his future.
A former Celtic youth player, the defender joined Newcastle United in 2016 but made only one senior appearance before being loaned to Kilmarnock in August 2017.
That move became permanent the following year when Findlay signed a three-year contract at Rugby Park.
He became a mainstay of the defence under former manager Steve Clarke and earned his first Scotland call-up in March 2019.
After Clarke was appointed national coach, Findlay made his international debut and scored in the 6-0 win against San Marino at Hampden Park in October 2019.
He is one of several Hearts signing targets for next season, although Tynecastle officials hope to do some business before the winter transfer window closes tonight.
They are seeking a new winger, midfielder and right-back if possible.