Parker's side were confirmed down after last week's 1-0 defeat to Manchester City at Turf Moor, ending a one-season top-flight stay. A decision on his future is expected imminently.
The Sun's Alan Nixon reports that Gerrard is a frontrunner for the vacancy.
"The respected figure lives in the North West and would jump at the chance of a challenge like Burnley," Nixon writes.
Gerrard, 45, has been out of management since leaving Saudi Pro League side Al-Ettifaq by mutual consent in January 2025. He turned down the opportunity to return to Rangers in October after the sacking of Russell Martin, and has since been working as a pundit for TNT Sports alongside Ally McCoist.
He spoke to talkSPORT earlier this month about his appetite to return to the dugout.
"I'm more ready than I have been. My family are back home in Liverpool, the kids are all settled at school. I'm all ears. But at the same time, I'm not chasing. I'm not searching, I'm not desperate. I'm in a good place.
"Work-life balance is fantastic at the moment but if the right opportunity comes, I'm all ears to listen to that. But I'm having a fantastic time at TNT with Ally and the boys and the girls, covering some big games."
Gerrard's last job in English football was at Aston Villa, where he was sacked in October 2022 after a poor start to the season. He guided Rangers to the Scottish Premiership title before that — the club's first in a decade — and reached the Europa League final in 2022.
Burnley will be keen to bounce straight back to the Premier League and view Gerrard as a high-profile appointment capable of galvanising the club. Parker won the Championship title in his first season before struggling to keep the club up. Whether the board retain him or move on will shape what is set to be a significant summer at Turf Moor.