The 26-year-old is entering the final 12 months of his current deal, with negotiations over an extension having stalled after 18 months of discussions.
However, Real have now presented fresh terms, which Vinicius is reportedly appreciative of, and the club is awaiting his decision, according to the BBC.
Senior figures, including returning head coach Jose Mourinho, are hopeful the Brazilian will agree to renew, with a response expected within hours rather than days. Should he reject the proposal, Arsenal are prepared to accelerate their pursuit.
Vinicius returned to Real on Monday for pre-season training and medical checks after the World Cup with Brazil. He has made 375 appearances and scored 128 goals for the club.
The winger reportedly wants improved financial recognition, while Real remain committed to their wage structure. Arsenal consider him their top left-wing target.