'The Gunners' are considering accepting Atletico Mineiro's 25 million euro price tag for the 20-year-old.
Bernard, a part of Brazil's triumphant Confederations Cup team last month, has also attracted interest from Chelsea, Tottenham, Porto and Shakhtar Donetsk.
Any deal involving the playmaker is likely to be delayed until after the second leg of the Copa Libertadores final between Atletico Mineiro and Paraguay's Olimpia July 24.
Rooney stays with Man U
Manchester United manager David Moyes stressed that Wayne Rooney is "not for sale".
"It's hard to keep repeating myself. But I will do. Wayne is a Manchester United player and will remain that. Unless I was speaking double Dutch I think we said Wayne Rooney is not for sale," he said Friday.
"I cannot speak about players from other teams but you know me. I have always said what I think and I like the player very much."
The BBC said that Rooney is travelling home from the club's pre-season tour in Thailand because of a hamstring problem.
Gold Cup tournament
Honduras got a second straight victory beating El Salvador 1-0 in a Group B match of the CONCACAF Gold Cup football tournament.
It was a close match as neither team was able to score in the first 90 minutes Friday. But Jorge Claros' shot from the right side of the box finally reached the goal in the last minute, sealing the win for Honduras.
In the other Group B match, Haiti defeated Trinidad and Tobago 2-0 to get the first win of the tournament.