The PA news agency understands the 28-year-old Republic of Ireland international is undergoing the first stages of a medical as the Magpies attempt to finalise a deal to take him to St James’ Park.
Hendrick formally left Turf Moor after four years when his deal expired on June 30 having failed to agree terms to extend his stay.
He joined the Clarets in a £10.5million (€11.6m) switch from Derby in 2016 weeks after starring in Ireland’s Euro 2016 campaign in France, where they reached the last 16 after a famous victory over Italy.
Hendrick made 27 appearances for Burnley last season, but none after the return from lockdown as negotiations stalled.
His capture would represent a start to the summer recruitment process for Magpies head coach Steve Bruce, who has spent the last few months wondering if he would even be in a job when the new campaign kicks off amid frenzied, and as yet fruitless, speculation over the club’s ownership.
Bruce is understood to have been handed a reduced summer transfer kitty of between £25million and £30m – the club smashed their transfer record with a £40m move for striker Joelinton 12 months ago – and whatever else he can generate in sales.
Newcastle are thought to be concentrating largely on English-based targets and Hendrick’s name appears on a list which is also thought to include the likes of Manchester United defender Chris Smalling, who spent last season on loan at Roma, Brentford striker Ollie Watkins and Bournemouth counterpart Callum Wilson among others.