Sky Sports television showed footage of the 25-year-old player boarding a private jet and said an announcement could come on Friday or at the weekend.
Gordon has had the best season of his career, netting 17 times for Newcastle across all competitions, including 10 goals from 12 games in the Champions League, and is in England's World Cup squad.
Gordon still has four years on his contract, according to media reports, after joining in January 2023 and signing a long-term extension in October 2024.
But Newcastle have failed to qualify for European competition next season, finishing 12th in the Premier League.
Newcastle manager Eddie Howe recognised this month that Gordon might have played his final game for the club after Bayern Munich expressed an interest, before Barcelona then swooped in.