According to Fabrizio Romano, the Spanish giants have offered £49million, which could rise to £58million with add-ons.
All parties are now waiting on Barcelona being able to register new players before a deal progresses.
The Leeds star has certainly been a wanted man this summer, with Chelsea also being keen on the player.
It was suggested that Thomas Tuchel's side were leading the chase to secure his signature, however a move to the Nou Camp now looks to be the most likely destination for the Brazilian star.
The 25-year-old moved to Elland Road in 2020 following the club's promotion to the Premier League under Marcelo Bielsa.
After scoring six goals and providing nine assists in his first campaign with the club, he found the net 11 times last term as he played a key role in helping the Whites avoid relegation from the top flight.
With a move to Barcelona looming, South American football expert Tim Vickery believes a move to the Nou Camp is a difficult proposition to turn down for any player from Brazil.
Speaking on White and Jordan, Vickery said: "If you're a Brazilian, and Barcelona come knocking, it's so hard to say no.
"In Raphinha's shoes, if it's not Barcelona, they might be a temptation just to stay put because of the World Cup.
"But it was clear to me that Leeds wouldn't stand in his way if he gets a suitable offer."