The Spanish striker surpassed Wilfried Bony as the most expensive recruit in club history last month after completing a £15.5m ($20.6m) move from Atletico Madrid, but was not available to make an immediate Premier League bow against Burnley and also missed subsequent fixtures against newly-promoted Hull City, League One outfit Peterborough United and defending champions Leicester due a thigh problem sustained during training.
Francesco Guidolin initially seemed uncertain on exactly when Borja would be available for selection, but expressed hope that he might feature after the international break.
Providing a latest fitness update via Swansea's official website on Friday, the manager revealed that the eagerly anticipated debut is likely to come at the Liberty Stadium on Sunday 11 September as title contenders Chelsea seek to preserve their 100% record under Antonio Conte.
"Borja has stepped up his recovery and has now rejoined the full group on the training field," he said. "He is progressing well and if things continue as they are I am confident he will be available for selection for the visit of Chelsea.''
Borja rose through the ranks at Atletico, but made just one senior appearance for the club against Getafe in 2013 before seeing his progress stunted by a serious knee injury. The 24-year-old subsequently spent time out on loan with Real Murcia, SD Huesca, Deportivo La Coruna and Real Zaragoza before netting 18 goals in 36 appearances for Basque La Liga outfit Eibar last season.
He joins compatriot Fernando Llorente in a revamped Swansea forward line that suffered the loss of Andre Ayew to West Ham United in a £20.5m deal. Portugal's Euro 2016 final hero Eder also departed on a permanent deal to Lille, Alberto Paloschi joined Atalanta and Bafetimbi Gomis was loaned to Marseille. The little-used Marvin Emnes linked up with Championship strugglers Blackburn Rovers on deadline day.
The Swans, who sold dependable captain Ashley Williams to Everton in early August, have lost two of their three top-flight matches so far this term and Guidolin will be under pressure to improve results after agreeing a new two-year contract to stay at the end of the 2015-16 campaign. In addition to Borja and Llorente, the experienced Italian brought Leroy Fer back to South Wales during the summer transfer window after a successful loan stint. Goalkeeper Mark Birighitti, Mike van der Hoorn and Alfie Mawson also arrived, along with U23 duo Tyler Reid and George Byers.