Burke went on loan to the Scottish champions when Brendan Rodgers was manager and started promisingly, scoring four goals in his first seven Scottish Premiership games.
But he barely featured under Lennon, and Dowling has now revealed he was unhappy with the treatment the 22-year-old received.
"He won’t be going nowhere near Celtic with the current manager," said Dowling. "They know what’s gone on and Oli certainly knows what’s gone on. It’s just something we’re not happy with at all."
Dowling did not wish to elaborate any more on the sort of treatment Burke received, but he was angry with how the forward's loan spell finished at Parkhead.
"We sent Oli to Celtic in good faith with Brendan Rodgers and Lee Congerton, some really good people up there that wanted to take Oli and wanted to play Oli," he said.
"They wanted to make him a better player than he currently is. We forget how young he still is.
"Once Brendan left to go to Leicester the treatment he got from the manager that’s in place now is something we don’t expect for one of our players. He won’t be going anywhere near Celtic."
Burke will be given a chance to prove himself to new Baggies boss Slaven Bilic.
"We currently have a new manager that’s excited to start working and looking at Oli," said Dowling.
"We believe he’s got an opportunity back here with a new manager in place.
"Through no fault of his own he’s been bought twice for £15m.
"If you look at the two fees and the amount of games he’s actually started - when when West Brom bought him look at the games he’d started, I think there was only 19 in his career. Nineteen games and you’ve been sold twice for £15m.
"There was big expectation when Oli came here. Naturally you’re going to get that when people pay that amount of money for you, but I don’t believe he really got the best chance to perform here.
"One minute he’s told he’s a right winger, next minute he’s a centre-forward then he couldn’t do that because he's told he couldn’t play with his back to goal.
"He’s had mixed messages and that’s why we thought going up to work with Brendan would be a good period for him to come back a better player and to improve on his game, but unfortunately that wasn’t to be. He needs stability."