During January, Sanchez completed a transfer to Manchester United, in a swap deal that saw Henrikh Mkhitaryan move to Arsenal.
United wrapped up a 2-0 victory against Huddersfield on a day when the clubs came together to mark the 60th anniversary of the Munich air disaster.
The winter window opens at one minute past midnight on January 1, but Barca are looking for a belated Christmas present in the France international.
Chile forward Sanchez stepped up to seal Arsenal's second successive league win with his fourth club goal of the season.
Hector Bellerin says Arsenal's 2-0 victory over Tottenham in the north London derby has silenced the doubters.
Shkodran Mustafi and Alexis Sanchez scored first-half goals to help the Gunners to restore a sense of supremacy in north London after a disastrous campaign last season.
Arsene Wenger has called on Alexis Sanchez to deliver on the pitch and silence those Arsenal fans who jeered him.
Alexis Sanchez is even more integral to Arsenal and embattled manager Wenger's hopes of silverware than ever before.
Manchester City manager Pep Guardiola personally made sure Raheem Sterling would not be allowed to join Arsenal.
For some of the other high-profile players returning to their English Premier League teams following the international break, the reception might be a little frostier.
Wenger has voiced confidence that Alexis Sanchez will put his aborted transfer to Manchester City behind him 'very quickly'.
Amid speculations about the Chilean's desire to leave for Manchester City, Arsene Wenger said he is confident of Sanchez's commitment.
Wenger faces a series of damaging transfer window defections that threaten to derail his side's already chaotic season.
For now, Wenger's focus remains on the trip to Stoke, where Arsenal won for the first time in seven attempts last May.
Wenger's admission will only heighten speculation the 28-year-old forward will join Premier League rivals Manchester City.