Newcastle were keen admirers of Abraham, who was on loan at Bristol City last year, but have missed out to Swansea City.
The Coventry man had been monitored by Leicester City, Brighton and Hove Albion and Newcastle United, but looks set to go to his native Wales.
Swansea City boss Paul Clement should call in a favour at Bayern Munich and ask to take Renato Sanches on loan.
The two players are key to Swansea, and while both have been linked with moves away, the club is keen to keep them in Wales.
Paul Clement's Swansea City could face competition from Spanish giants Valencia, however.
Swansea City will be looking to bolster their attacking options this summer to restore themselves to the top half of the Premier League table.
The Derby academy graduate left for West Bromwich Albion in 2010, and is now at MLS side Orlando City.
Paul Clement believes Gylfi Sigurdsson is better off being the key player at Swansea City rather than moving on this summer and risking a repeat of his experience at Tottenham Hotspur, where the midfielder spent two years trying to hold down a regular first-team place.
Matt Grimes and Kyle Bartley played on loan at Leeds United from Swansea City in 2016-17.
Swansea City boss Paul Clement should enter the race to sign Chelsea forward Tammy Abraham, and his previous ties with the club could give the Swans an advantage if they do bid for him.
When Paul Clement was appointed in January by Swansea City, one of the questions at his first press conference was framed around the club’s dire predicament, specifically what made the third manager to take charge in as many months think that relegation was not a formality for a team anchored to the bottom of the table with the worst defensive record in the division.
As Paul Clement talks about having played Saturday’s pivotal game at Sunderland 100 times in his mind already, it is clear that moments of escapism are rare for the Swansea City manager, although he did find something other than the Premier League relegation battle to occupy his thoughts for an hour or so this week.
Paul Clement believes keeping Swansea City in the Premier League this season would be his greatest achievement in football, eclipsing anything he achieved during his time as assistant manager at Real Madrid, Paris Saint-Germain and Chelsea, where he enjoyed huge success alongside Carlo Ancelotti.