Kompany's side travel to Old Trafford on Saturday hoping to boost their survival hopes by inflicting United's ninth home defeat in all competitions this season.
Losing to second-bottom Burnley would be a significant blow to Erik ten Hag's hopes of staying on as United manager beyond this season.
United are languishing in sixth place and Ten Hag's position is under scrutiny after British billionaire Jim Ratcliffe's recent arrival as co-owner.
United needed penalties to see off Coventry in the FA Cup semi-finals after blowing a 3-0 lead and twice trailed bottom-of-the-table Sheffield United before ending a four-game winless league run with a 4-2 victory on Wednesday.
During Kompany's 11 years as a lynchpin of City's golden era of success, the Belgian often had to battle against dominant United teams.
That is no longer the case but he is not relishing the Old Trafford club's demise.
"Pleasure? No, come on," he said. "On the other side is a manager that I have a lot of respect for, that I know the quality of, that I've seen and faced in different leagues.
"He was at Ajax when I was in Belgium with Anderlecht, I've seen the quality of his teams and what he's been doing, so on the contrary, there's a form of respect there, but I'm more about our business.
"Of course I'm happy and I can't hide it if City do well but my whole world exists of Burnley and what we can achieve."
Kompany, a four-time Premier League champion with City, has plenty of happy memories of visits to United as a player, not least their famous 6-1 win in 2011.
But he isn't dwelling on past successes with Burnley three points from safety and just four games left to avoid an immediate return to the Championship.
"I want to build memories as a manager at Old Trafford. I put playing memories aside for now," Kompany said.
"It's a meaningful game. It's been a meaningful game for many years in my life but it's really about Burnley and Burnley facing United on the day.
"It's for this club where I want to be successful there on Saturday."
