One newspaper report claimed Villa had earmarked just £1m for new players, after heavy spending failed to earn them promotion to the Premier League.
Wyness did not reveal how much they would have to spend but said boss Steve Bruce understood the club's position.
"There are about three or four targets," he told BBC WM.
"We can easily achieve that with the budgets we have."
Villa spent £39m alone on strikers Jonathan Kodjia, Ross McCormack and Scott Hogan following their relegation from the top flight in 2015-16.
Wales defender James Chester, Albert Adomah, Mile Jedinak, Conor Hourihane, Henri Lansbury and Tommy Elphick were among 12 other additions.
Wyness dismissed reports that Bruce, who took over at Villa Park in October and guided the club to 13th in the Championship, had been surprised by his budget for next season.
"I've spoken with Steve and the words 'shocked' and 'stunned' didn't come to mind - [it's] maybe journalistic speak. Steve understands completely," he added.
"I'm already in discussions and active on a number of deals."
Former Aberdeen and Everton chief executive Wyness, who joined Villa last summer under new owner Dr Tony Xia, said the club's hierarchy was aware performances needed to improve.
"There's no getting away from it, this summer is a key summer for Villa," he said.
"Thirteenth is just not acceptable, we know that. I think it's below a pass. We didn't achieve anywhere near the targets we set out for.
"However, 'must do better but there are promising signs' - I think that's the report card."
Chief executive Keith Wyness says Aston Villa remain confident of making three or four signings this summer with the budget the Championship club has set.