The 19-year-old agreed to join the Swans shortly after penning a new contract at Stamford Bridge, but his move to the South Wales outfit has been delayed because he was on holiday after helping England’s Under-21 side reach the semi-finals of the European Under-21 Championship in Poland.
However, Abraham, who scored 26 goals on loan at Championship side Bristol City last term, has now joined up with head coach Paul Clement and the rest of the squad on their United States tour.
The Swans are now in North Carolina having travelled there from Philadelphia following a 2-2 draw against MLS side the Union at Talen Energy Stadium.
The next game is on Wednesday against Richmond Kickers and it is likely that Abraham will get his first taste of action in a Swansea shirt, before the second and final game of the trip against North Carolina.
A goal apiece from James Tomkins and Luka Milivojevic handed Crystal Palace a crucial 2-0 win over Huddersfield at the John Smith's Stadium on Saturday evening.
Manchester United bounced back from their Champions League exit as they beat Brighton 2-0 at Old Trafford on Saturday night to reach the FA Cup semi-finals.
Tottenham reached the FA Cup semi-finals after they cruised to a 3-0 victory over Swansea in their quarter-final clash at the Liberty Stadium on Saturday.
Premier League basement club West Brom squandered the lead as they suffered a 2-1 defeat to Bournemouth at the Vitality Stadium on Saturday.
Chesterfield boss Jack Lester felt his side were unlucky not to claim all three points after their 1-1 draw at Cheltenham.
New Swindon manager Phil Brown revelled in his side's victory at Cambridge in his first game in charge.