The two clubs reportedly sent scouts to watch the Spanish under 21 international in November and December.
After an impressive start to the 2023-24 season, an ankle injury then kept him out for three weeks before making his return almost a fortnight ago.
In all competitions, he has registered four goals and two assists in 19 starts for the La Liga side.
Arsenal are hoping that Barrenetxea will provide competition for Gabriel Martinelli and Bukayo Saka.
Meanwhile, Spurs boss Ange Postecoglou likens the 22-year-old to the winger he previously coached at Celtic, Jota.
They have already signed Timo Werner in this window but another attacking option would still be welcomed.
How good is Ander Barrenetxea?
Barrenetxea came through the academy at Real Sociedad and became the first player born in the 21st century to play in La Liga when he made his debut against Alaves in 2018.
He was also the first player to skip the B team in his route to the senior squad since Antoine Griezmann.
— Real Sociedad 🇺🇸 🇬🇧 (@RealSociedadEN) January 18, 2024
Yet, despite all of the promising signs, Barrenetxea has never really exploded onto the scene like a player of his talent is expected to.
He has made north of 120 appearances for the club and only managed 13 goals and six assists.
However, he has a knack for producing stunning moments and these moments are becoming more frequent this season.
Barrnetxea put in a stunning performance against Benfica in the Champions League, scoring a brilliant goal and setting up a penalty (that was missed by Brais Mendez) with an outrageous piece of skill.