Pulis: Jones took pay cut to join Potters
It is understood Jones was earning around £60,000 per week in his final contract with the Wearsiders, way beyond Stoke's financial clout.
But as well as agreeing to spread the £8 million paid for Jones over a period of four years, Sunderland have also assisted Stoke with regard to the wages.
Pulis refused to reveal details, although it is believed Jones was given a one-off goodbye payment from Sunderland, easing him off their wage bill.
The Trinidad & Tobago international now falls into line with the wage structure at the Britannia Stadium, a crucial element in the deal as far as Pulis was concerned.
"The wages he was earning at Sunderland we could not afford. He was on quite a bit of money, but he is not on that at Stoke,'' Pulis said.
"So they have looked after that as well to enable him to come down to the structure we have at this club, which is very important for us.
"We want to keep it as tight as we can, and we've been able to do that. It was as big and an important a factor as the £8million spread over four years.
"It's a point we have to make because I don't want anybody within the club or from outside thinking they can walk in here and their demands are going to be this, that and the other.
"We work very hard on the structure at this club, especially wages, and there was no way we could do the deal unless it conformed with those.''
Key for Pulis now is unlocking Jones' potential week in week out as his inconsistency was a cause of frustration for Sunderland manager Steve Bruce.
"I have heard Steve's interview, and he does have the inconsistency about him,'' Pulis said. "There were times at Sunderland when he was absolutely outstanding, unplayable, but there were other times when you watched him and you wondered whether it was the same player.
"So it's a question of whether we can get that little bit more out of him, whether he can mature with us and learn from what happened at Sunderland.
"Sometimes you need a fresh start and it might be Kenwyne needs a fresh start, but also we need Kenwyne's help as well. He has to help himself first and foremost.''
Pulis, meanwhile, is poised to offload James Beattie to Scottish Premier League champions Rangers. The former Blackburn, Southampton, Everton and Sheffield United forward was involved in a much-publicised bust-up with Pulis in early December last year.
Despite suffering with recurring fitness issues, Beattie rarely figured over the second half of the season.
With Jones' arrival, Pulis now has the opportunity to move unwanted players on, and Beattie looks set to leave.
"James is up talking to Rangers, so I don't know how that is going on at the moment,'' Pulis said. "If it goes through it will be a permanent deal.''
With cash still to spend given the deal for Jones, Pulis is hoping to tie up a move for West Ham midfielder Radoslav Kovac. Czech Republic international Kovac only joined the Hammers a year ago from Spartak Moscow for £2 million.
But with Salif Diao opting not to sign a new deal with the Potters, much to Pulis' disappointment, the City boss is now determined to strengthen the midfield area.
"It's now down to West Ham. The ball is in their court,'' Pulis said. "It's up to them to decide whether they want to do a deal or not.
"There's been a bit of too-ing and fro-ing. One minute it's yes, the next they're not sure, so we will leave it to them.''