The Bluebirds have already broken their transfer record twice this summer on Andreas Cornelius and Steven Caulker, and their total spend is already nearly at the £20million mark.
But Mackay is hopeful of adding up to four more new faces, with Toulouse's Etienne Capoue one of those he has made a bid for.
"We still have three or four irons in the fire and we have until September 2nd [when the transfer window closes]," Mackay told Sky Sports as part of our 92Live.
"We may well start the season still chasing targets but by the end of August I hope to have the squad together than can absolutely compete."
Mackay is determined to make sure Cardiff are ready to compete in what will be their first top-flight campaign in nearly half a century.
"We have got to make sure we are very competitive, we managed to get here for the first time in 51 years and we have to make sure we stay here. The fans have been fantastic and it is about us bringing in players that can challenge everyone," he said.
"No doubts it is going to be tough but we have to make sure we are competitive and we have made sure we have signed a couple of players already."
Mackay was full of praise for his owner Vincent Tan, who has backed him with major funding.
"He has spent £10million on the training ground, he dealt with club debt last week and he has allowed me to bring these players in. It is about progressing the club and for that you need someone at the top who has got vision," he said.
"There is transparency there, whatever he wants to say he says direct to me and when I have messages about the club, it goes direct to him and not through three or four lines.
"The backing he has given me has been fantastic, and we want to progress and compete at the highest level. To do that we need higher quality players, we did that with Cornelius, and then with Steven.
"I have been chasing him for two years but when the chance came up I wanted to make sure we could talk to him and I am delighted that he has committed to us, he is in the England squad and I think he could be England's future at centre-half."