Kompany, officially announced as Sean Dyche’s replacement three weeks ago, had his first success in the market last weekend when landing MK Dons’ forward Scott Twine for £4m.
The League One player of the year can play off the front, wide or through the middle, and scored the only goal in Friday’s opening friendly victory against Rochdale at Gawthorpe.
The Belgian then added centre back Luke McNally to the mix, arriving from Oxford United for £2m, before returning to former club Manchester City to snap up young defender CJ Egan-Riley in a deal that could rise to £1m, and then England Under 21 skipper Taylor Harwood-Bellis on a season-long loan.
Kompany looked at the state of play, and how he is enjoying working with his signings: "Every day I keep discovering the players we have now.
"We will keep looking at the market and see how we can strengthen, but at the same time, my first idea is to look within our team and plug the gaps where we need to.
"There is a good vibe about it, so we want to maintain that and see how we can strengthen the team."
He is impressed with what he has seen from Twine, and feels there is great potential there: "It starts with the attitude, mentality and work-rate, and he is a talented young player.
"He is really intelligent, plays in between the lines and also runs in behind.
"In the box he is lethal, but we are giving him time. He needs to adapt to the physicality that we will face.
"But we are happy to have him in the building and look forward to see him grow to his maximum level."
McNally only spent 18 months at Oxford after arriving from Ireland, and Kompany noted: "Luke is a player who has a good mix – he has had a steep progression.
"Everything in his career has been a constant step up, and he looks unfazed.
"This is the highest he has been, and he looks unfazed.
"He has stepped into it quite nicely, and physically he is a presence and he is calm on the ball.
"He is one of these players we think we will get a lot of joy from in the future."
Burnley had the oldest squad in terms of average age in the Premier League last season, but with Twine and McNally both 22, Kompany has also added Egan-Riley at 19, and Harwoodf-Bellis at 20.
On Egan-Riley, he said: "That is another one we look to complete to have a good balance.
"We have a very, very good core of experienced players, and someone like CJ coming in freshens it up a little bit.
"He can play in a number of positions, but I think his potential as a centre back in the modern game is interesting.
"It starts with the attitude, and he has plenty of very good habits for our club, and the rest will be time and we will support him to become the best version that he can be."
Harwood-Bellis spent the first half of last season with Kompany on loan at Anderlecht: “Taylor is a player that has a lot of experience already at this level, more than 70 games in the Championship, played in Belgium, captain of England Under 21s and played at the highest level with City.
"But the biggest thing with this kid is he has an incredible work-rate and attitude to the game.
"He will help, not just as a talented player, but also as a leader in our team.
"We need those characters after losing guys like Ben Mee and (James) Tarkowski, who were really good players, but also characters in the dressing room."