Roma are claimed to have slapped a €100m asking price on Alisson's head, but Taffarel, who won over 100 caps for Brazil, revealed the goalkeeper told him he would be happy to remain in the Italian capital.
Alisson wants future cleared prior to the World Cup with Liverpool looking the most likely destination for the Brazillian shot-stopper.
Real Madrid are also interested in Alisson, and although the player has said he'd like his future resolved before Brazil's opening game of the World Cup against Switzerland on Sunday, this seems very unlikely. The truth is that I left everything in the hands of my manager.
Alisson shone last season between the sticks for Roma and has been eyed by Liverpool, Real Madrid and Paris Saint-Germain in the current transfer window.
He told UOL Esporte: 'I think everything will be resolved before the start of the World Cup, that's what I want. "We'll see if something happens during this week".
Liverpool are reportedly keen on signing the Roma goalkeeper.
Brazilian outlet Terra claim Liverpool have launched a R$285m offer for the 25-year-old, which works out at £57m.
Alisson's performances this season have made him one of the most in-demand goalkeepers in Europe, something which will only increase further following the World Cup where he is expected to start for Brazil ahead of Manchester City's Ederson.