Bayern Munich have been leading the race to sign him this summer, and have an agreement in place on personal terms, but Barca have now matched that offer.
Gordon had been leaning towards a move to Bayern, who are not giving up at this stage, but there is a feeling he now sees Barcelona as his preference.
Chelsea, Liverpool and Arsenal have all shown interest in keeping the player in the Premier League, but a move abroad has begun to look very likely.
A deal worth around £80 million would be needed to take Gordon away from Newcastle, and that is now considered to be a hurdle that all interested parties have generally accepted.
Barca have a slightly different view of his valuation, but talks over that are now progressing.
Flashscore sources indicate that talks around the potential deal for Gordon to move to Barcelona initially began two weeks ago and accelerated quickly once they realised they could beat rival clubs to the deal.
Gordon has been with Newcastle since 2023, when he signed from Everton in a deal worth £45 million.
The 25-year-old is capable of playing across the front line and has won a place in England's World Cup squad, producing 17 goals and five assists across last season.
His primary role has been on the left flank, and the situation with Barca may yet have a knock-on effect for Marcus Rashford, who has been on loan at the Spanish club and is hoping to complete a permanent switch from Manchester United this summer.
There has been a general feeling that his time at Old Trafford is over, but a deal for Gordon would bring that into some doubt because of the players' positions and the financial obligations involved.