According to Portuguese newspaper A Bola, Monaco made the approach during negotiations over Leonadro Jardim's move from Sporting to the Ligue 1 team. Jardim is set to replace Claudio Ranieri as Monaco boss from next season.
But when they asked what Sporting's asking price would be for the Portugal international, Sporting said anything less than Carvalho's €45m release clause would not be considered.
Manchester United and Liverpool have both been linked with Carvalho, although neither Premier League side wants to pay the full release clause.
Reports suggested Liverpool were willing to pay up to £30m for the 22-year-old midfielder, who is set to play for Portugal at the World Cup this summer.
Carvalho was named in the 30-man squad for the Finals, along with teammate Rui Patricio. Cristiano Ronaldo captains the side leads the attack.
United have a strong track record when it comes to signing young Portuguese players from Sporting, having snapped up Nani and Ronaldo in the past.
Liverpool meanwhile are also expected to strengthen significantly over the summer as they prepare for a Champions League campaign next season.
And while both United and Liverpool would likely baulk at Sporting's demands, it's encouraging for both teams that Monaco have dropped out of the race.