The Independent says the 22-year-old is one of the top transfer targets of new Barcelona boss Ernesto Valverde, with the Catalan club hoping to agree a deal with the Gunners in the region of £45m ($57m).
However, Arsenal are standing firm in the knowledge that Barcelona's interest in Paris Saint-Germain midfielder Marco Verratti could put a deal for Bellerin out of the financial reach of the La Liga side.
Self-imposed budget restrictions mean Barcelona can only afford to spend around £75m net this summer, with a deal for Verratti likely to take a significant chunk out of that total.
The Gunners are wary of losing one of their prized possessions, especially with the future of star forward Alexis Sanchez also up in the air.
Bellering signed a six-and-a-half year contract worth £95,000 a week last November.
Barcelona midfielder Denis Suarez said last week that Bellerin wanted to join Barcelona as he had his family in the city but that he felt tied down to Arsenal.
"I have spoken to Bellerin but it is not an easy situation because Arsenal trusted in him since he was a boy," Suarez was quoted as saying by the London Evening Standard.
"It is his decision, and Arsenal's decision. I really don't know what is going to happen but I wish he can come.
"I have spoken to him and from one side he wants to join Barca, but from the other he feels very attached to Arsenal. It is a hard situation and I am nobody to be speaking about this, but I wish he could come."
But Bellerin's teammate Nacho Monreal dismissed speculation linking the right-back with Barcelona and insisted that he will still be at the club next season.
"He's an Arsenal player and almost certainly he will stay here," he was quoted as saying by the Daily Mail last week.