The Spanish publication says that the La Liga giants want to reward the France centre-back for his impressive performances but also plan to increase his release clause to over €200m (£176m, $250m) following the recent speculation linking him with both Manchester United and especially Manchester City.
The 24-year-old defender joined Barcelona from Lyon during the summer of 2016 in a deal worth €25m.
The Frenchman was tipped to become the long-term replacement for Javier Mascherano and finally fill the big shoes of legend Carlos Puyol, but his arrival sparked some initial doubts following the previous failures of Jeremy Mathieu and Thomas Vermaelen.
But Umtiti has since developed into of the best centre-backs around Europe, establishing himself as a key player of the Barcelona back-line alongside Gerard Pique.
His future at the Nou Camp consequently came under some scrutiny during the recent January transfer window with his current €60m release clause looking like a bargain for a player of his age and potential.
The Daily Mail claimed both City and United were closely monitoring the situation with Pep Guardiola then on the hunt for a new centre-back to bolster his defence.
L'Equipe then reported that the City boss was considering making the move in January - before the Premier League leaders decided to instead trigger Aymeric Laporte's €65m clause at Athletic Club Bilbao.
Sport says that Umtiti has told Barcelona officials that he is happy at the Nou Camp and the club are not concerned over losing him even if a club try to meet his current clause.
But the La Liga giants still want to give him a pay rise and extend his contract until 2023 as the France international is currently the starter who earns the least in Ernesto Valverde's side.
Captain Andres Iniesta and Lionel Messi inked new deals at Barcelona earlier this season and last month Gerard Pique and Sergi Roberto followed the same path by agreeing new contracts which both included mammoth €500m buyout clauses.
Umtiti is expected to be the next one and Sport says that the idea is to get a deal done with a release clause worth more than €200m before the upcoming Summer World Cup.
The Spanish publication says that the club are confident that the negotiations will be straightforward after Valverde last month expressed his hopes of keeping him at the Nou Camp for the long-term.
"I don't have to worry about what other clubs could do. It is clear that Umtiti is a great player and we hope we can continue to enjoy him here for many more years," Valverde said when asked whether he was concerned about a club trying to pay his €60m clause.
"This depends on the legislation. Here players have clauses and each player has a price. That's how the legislation works here. Those release clauses can seem really high and then, two or three years later, it doesn't seem that much. Anyway, don't forget that for a player to leave by paying his release clause the player has to want to leave, too. But we're really happy with Umtiti and we think that he is happy in Barcelona and that he'll be here for a long time."