BBC Sport Wales also understands Arthur has apologised to Cardiff City for similar remark about Peter Whittingham.
"I have spoken to Mark and he's made his apologies to us," said Swansea chairman Huw Jenkins.
However, Jenkins has called on the sport's authorities to sort out the issue of "tapping up".
He said: "I think in football in this day and age there's a lot of conversations going on between agents behind the scenes with players and other clubs.
"And putting our Darren Pratley to one side I think the Football League does need to get a grip of it and sort things out."
Cardiff City and Swansea City have already said they are to urge the Football League to investigate Arthur for trying to unsettle their star players.
Arthur said Pratley and Cardiff's Peter Whittingham want to sign for Forest.
"If that's not blatant tapping-up then I don't know what is," insisted Cardiff boss Dave Jones hitting back. "The league need to have a look at it."
And Swans chairman Huw Jenkins added: "I think it'll be worth looking at."
Forest have had bids rejected for both Bluebirds winger Whittingham, last season's Championship top-scorer, and Swansea's key man Pratley who are both in the final year of their contracts at their respective clubs.
And Arthur who boasted Forest are "only one of a few clubs with any money to spend".
Forest accused of 'tapping up' Whittingham
It has sparked a strong response from Wales' two Championship clubs.
"It is not right, it is not within the rules and laws of the game," said Jenkins.
"And I think we should look at it a bit further. Perhaps ourselves and Cardiff can team up and protect our clubs a little bit."
Cardiff insist they do not need to sell their star players such as Whittingham, who was also subject of a failed bid from Championship rivals Burnley in June, and Michael Chopra despite their estimated debt of between £15m and £30.
But Arthur said about Whittingham: "We are aware he is in his last year of his contract and I'm sure they're not in a financial position where they can afford to let that run down.
"We understand he has rejected a new offer so we have to be a little bit patient and hopefully between now and the end of August we'll be able to bring him in.
"If not, then we'll try to bring him in either in January or at the end of the season when he is out of contract because we know he wants to come to us."
Cardiff reacted angrily as chief executive Gethin Jenkins released a statement that read: "I would like to thank Mr Arthur for his interest in our football club though I would suggest that he direct his attention towards internal Nottingham Forest issues and problems than any blatant public attempts to unsettle members of our playing squad."
Jones was equally furious as Arthur's comments came on the eve of the new Football League season and while former England under-21 international Whittingham, who scored 25 goals last season, is in the middle of contract negotiations with Cardiff.
"He has no right to say it," said Jones. "And he should go though the proper channels to say it rather than through the media, spouting off that our player is unhappy.
"If he is talking to Whitts on that score, it must be his brother because that is not Whitts is telling me.
"We've put an offer to Peter Whittingham and his stance is he wants to see what happens here, whether the club wants to move forward.
"Now he has those reassurances, now we can go back to him and can start the negotiations again."
Asked if Forest should be reported to the Football League, Jones replied: "I don't know what you can do.
"But how can you you talk about another team's players, the club finances and saying that "he knows he wants to come to us"?
"If the roles were reversed, how would they feel about us blatantly trying to upset a player? They either put up or shut up and let us get on with our business."
Forest have made just one signing so far this summer, Chelsea left-back Ryan Bertrand agreeing a six-month loan deal and Arthur is keen to sign Whittingham and Pratley before the transfer window closes on 31 August.
Former Fulham trainee Pratley, the target of a failed January bid by Forest, has been an integral part in Swansea's two narrowly unsuccessful Championship play-off bids in the last two seasons.
Cardiff chief exec Gethin Jenkins 'unhappy' with Forest chief Arthur
And new Swans boss Bendan Rodgers will be loathe to lose 25-year-old Pratley so soon after fellow midfielder Leon Britton left on a free transfer to Sheffield United.
"They've offered him a new contract and apparently he has turned it down," said Arthur. "Again we understand he is very keen to come and play for us."
But Jenkins was "surprised" and "disappointed" about Arthur's up-front comments about Pratley wanting to leave Swansea and sign for Forest.
"It is an absolutely astonishing statement to make by somebody in Mark's position," said the Swans chairman.
"They are obviously not getting the answers they want and are trying any tactic they want to try and get what they want which is very, very disappointing.
"But we are still talking to Darren and his agent about extending his contract."
Source: BBC Sport
Source: BBC Sport