The British Olympic football squad member's contract is due to expire next summer, and the 23-year-old - who scored in the 5-0 win over QPR at the weekend - has informed Swansea chairman Huw Jenkins and manager Michael Laudrup he has no desire to sign a new contract with the club.
City have been regularly linked with a move for the former Chelsea midfielder, and are reported to have firmed up their interest with a bid of around £5m (Dh29m).
"He's not really interested in signing a new contract and I think Man City have made a bid," Swansea coach Alan Curtis told BBC Sport.
"So it's just a matter of time now before it happens and it's good for both parties."
The 28-year-old's status as regular starter at Spurs was placed in doubt after he was left out of the squad for Saturday's opening day Premier League defeat at Newcastle.
QPR are in the market for another centre-back, and see Dawson - who made just 13 appearances for Spurs last season through injury - as the right man to lead the back line.
Finland international captain Niklas Moisander is set to return to Ajax after the Dutch giants agreed a EU3m (Dh13.6m) deal with AZ Alkmaar.
The 26-year-old, who is seen as a replacement for Jan Vertonghen, spent three years in the Ajax reserves after joining them as a 19-year-old from TPS.
He moved to FC Zwolle in 2006 before signing for Alkmaar two years later, eventually becoming their captain.
He returns to the Amsterdam Arena on a three-year deal.
Poland international midfielder Dariusz Dudka has joined Spanish side Levante on a free transfer following Auxerre's relegation last season.
The versatile Dudka, who has 65 caps for Poland, said: "I chose Levante because they gave me the best offer.
"Russian and Turkish clubs were also interested in me, but I believe I can perform the best in Spain."
Liverpool defender Martin Skrtel has ended speculation about his future after agreeing a new long-term deal with the English Premier League side.
But the 27-year-old Slovakian stopper - who joined Liverpool in 2008 - will remain at the club after agreeing a 'long-term' contract.
"After last season there was a lot of talk about my future but I have to say that the first priority for me was always to sign a new deal with Liverpool," he said.
The Cameroon international successfully passed his medical with the Spanish side and put pen to paper on a five-year contract, which has an EU80m (Dh363m) buy-out clause.
French defender Yohan Benalouane is to join Parma on loan from Cesena.
The 25-year-old, who had been linked with a move to Newcastle in the summer, will remain in Serie A despite Cesena's relegation - with Parma considering acquiring a 50% stake in the player's ownership next season.