Flanagan is in the final season of his three-year Bulldogs contract but was dropped to reserve grade last week to make way for the double act of veteran Reynolds and Oloapu.
The 24-year-old has again missed out on selection for round 10 and has been linked with a possible move to Manly, where Josh Schuster has had an unconvincing start to life as a five-eighth and father Shane Flanagan is an assistant coach.
On social media on Wednesday, Bulldogs general manager of football Phil Gould strenuously denied such a move was in the works, with Burton hopeful Flanagan would remain at the club well into the future.
"There's obviously a lot of talk and speculation around it but we'd love to have Kyle here in the long-term," Burton said.
"We see him as a great teammate."
Flanagan recovered from a similar axing in 2022, returning to the first-grade side after starting the year in the reserves, and played hooker for Canterbury's NSW Cup side on the weekend.
Burton switched to centre in Penrith's run to the 2021 grand final so he knows first hand that utility value could strengthen Flanagan's case for an NRL recall.
"It's obviously good to have a few positions under your belt. I've obviously had the opportunity to do that with centre," he said.
"If he works on that, he can add that (string) to his bow. That'd be great."
