Hendrick, who has been capped 54 times by the Republic of Ireland, is out of contract this summer and has yet to agree with the Premier League side to extend.
Before the coronavirus pandemic, Milan indicated their interest in Hendrick and are currently reviewing their options ahead of the next transfer window.
The Series A giants — like their Inter rivals — have looked more and more at the English market in recent months and came close to signing up in January to Wigan defender Antonee Robinson.
Burnley had initial talks on a new contract with Hendrick and remained hopeful to keep him at Turf Moor.
Several other Premier League clubs expressed interest in Hendrick, who was a big player in the midfield of Burnley under manager Sean Dyche.
The 28-year-old joined Burnley from Derby County for a £10.5 m club record fee at the time in August 2016 and has made 139 appearances for the club, scoring 10 goals since.
Hendrick is back in training with Burnley as the Premier League gears up to June 17.
When the Premier League resumes, Dyche ‘s men are still racing for a Europa League spot. Signed from Derby County in 2016 for £10.5million, Hendrick has clocked up Burnley’s 107 Prem starts.
Yet he’s now thought to be weighing his options and at 28, might be fantasizing his Italian work in what may be his last long, big deal.
Ramsey is at Juventus, whereas Young moved in the January Transfer window to Inter Milan.
Milan had been involved in getting Hendrick to the San Siro since the coronavirus pandemic exploded, according to Sky Sports.