The 24-year-old is currently spending the second half of the 2023-24 campaign on loan at Ashton Gate, following an agreement between the West Country club and the Clarets in the January window.
Twine was at fellow Championship side Hull City during the first half of the term, however his stay with Liam Rosenior's Tigers was cut short early.
Since arriving at Bristol City, the attacker's game time has suffered massively due to a recurring quad injury, limiting him to just 10 appearances for the Robins.
The former MK Dons man lined up on Saturday afternoon against already-relegated Rotherham United, curling in a brilliant free-kick to help his side to a 2-0 win over the Millers.
Burnley's Twine wanted permanently by Bristol City
Speaking after his team's comfortable victory over the Championship's whipping boys, Bristol City boss Manning provided some insight into Twine's situation ahead of the off-season trading point.
"It's tough, ultimately he is not our player. It is one of those where the discussions have to happen for me to see what it looks like, whether it is possible," Manning told the Bristol Post following the beating of Rotherham.
"Burnley might not want to do it, there are so many permutations to it that - at the minute, I have no idea.
"I would love to work with him. I think he fits the club and profile of player we have got but there are so many challenges that go into it.
"For me, he has been great; I think he has got another level he can go to, I think he has starting to show that and he fits what we are trying to do."
State of play for Twine at Burnley
Although Twine is contracted at Bristol City on a temporary deal until the end of the campaign, there is no option or obligation to buy the 24-year-old during the summer transfer window.
Therefore, the Robins and Burnley will need to agree a transfer fee for the attacker if he is to move permanently to Ashton Gate, with the Clarets said to be valuing him at £5m prior to the January window.
One of the "many permutations" Manning refers to is whether Burnley will be operating in the Premier League or the Championship alongside his side next term.
Following a spirited display and 1-1 draw at Manchester United on Saturday afternoon, Vincent Kompany's troops are two points behind 17th-placed Nottingham Forest.