Abraham scored 26 goals for Bristol City in the Sky Bet Championship last season and Swansea chairman Huw Jenkins says the Welsh club are battling Premier League rivals Newcastle and Brighton for his signature.
Chelsea are keen for Abraham to gain more experience on loan, and the 19-year-old knows Paul Clement from the Swansea manager’s time as a coach at Stamford Bridge.
“Tammy does not have Premier League experience, but we are interested,” Jenkins told BBC Wales. “Paul knows him from Chelsea. If you look at the options which have been publicised, (they are) Brighton, Newcastle and ourselves.
“So it is left to Tammy and his family to decide where he wants to go next.”
Abraham is currently with the England Under-21 squad at the European Championship in Poland. He said last week that he would wait until after Euro 2017 before deciding his next move.
“I’m not too sure (about his future). Hopefully it will be in the Premier League,” Abraham said. “After this season I need to take myself to the next level. That’s the type of person I am, I always want to do better.”