The Scottish international was signed by Brendan Rodgers in January on a loan deal.
Burke scored four goals in 19 games for the Scottish champions, and played eight times under Neil Lennon.
However, West Brom took exception to the Northern Irisman's handling of their player.
“Oli Burke will not be going anywhere near Celtic with the current manager,” said Dowling. “They (Celtic) will know what went on. Oli obviously knows what happened. It is a mutual decision that he will never return to Celtic.
“We sent Oli to Celtic in good faith with Brendan Rodgers and (head of recruitment) Lee Congerton – really good people up there. They wanted Oli and I am sure they would have made him a better player.
“Once Brendan left to go to Leicester, the treatment he got from the manager in place – well, subsequently we do not expect any of our players to go anywhere near Celtic Football Club. We currently have a new manager who will start working and looking at Oli and hopefully he now has an opportunity here at Albion.”
Celtic have responded with an official statement.
"These comments, which question the professionalism of both Neil and Celtic Football Club, are ridiculous and completely inaccurate. West Brom have made no contact whatsoever with either Neil or the club in relation to Oliver.
"As with every player who comes to Celtic, Oliver received the very best of treatment and every support and opportunity. We wish Oliver the very best for the future."
Dowling revealed that the club wanted him to work with Rodgers after a difficult time at the Baggies.
“Through no fault of his own, he has been bought twice for around £15m,” he said. “If you look at the two fees and the number of games he has started it is a strange situation. He only played in 19 games and was twice sold for £15m.
“There were big expectations when he joined Albion. Naturally you are going to get these expectations after paying that amount of money.
“But I do not believe he really got the best chance to perform at Albion. One minute he was told he was a winger and the next minute he was told he was a centre forward.
“He had mixed messages and that is why we thought it would be a good period for him to work with Brendan and return to us a better player but unfortunately that wasn’t to be.
“He is now a player who needs some stability to show his real form.”