As reported by Portuguese newspaper Diário de Notícias, Monaco - who will be under the management of former Sporting Lisbon coach Leonardo Jardim next season following the departure of Claudio Ranieri - submitted a proposal of approximately €33 million for the player that was not accepted.
The source reports that Carvalho's current release clause at the Estadio Jose Alvalade stands at around €45 million, with Monaco's offer obviously falling someway to short of that number.
Premier League rivals Manchester United and Liverpool will presumably be buoyed by the news that the Ligue 1 outfit have been so far unable to convince Sporting into parting with the Portuguese international starlet, with reports last month having speculated that the duo were losing ground in the race to sign Carvalho.
Manchester United are expected to strengthen heavily in midfield this summer, with Carvalho - who will appear at the World Cup this summer as part of Paulo Bento's final 23-man squad for the tournament in Brazil - possibly viewed as an exciting long-term option under new boss Louis van Gaal despite his relative lack of experience at the highest level.
A robust defensive presence who is also capable in possession, the 22-year-old enjoyed something of a breakthrough season in Lisbon during 2013/14 and it is hardly surprising that Jardim is keen to see Carvalho continue such a quick progression under his leadership.
Liverpool boss Brendan Rodgers - who earned plaudits from across the footballing community having kept the Reds in the title race until the final day of the season despite relatively modest expectations - is also known as a coach who is more than willing to give young players the chance to thrive in the first team.
With Liverpool also believed to be in the market for several key signings over the next few months, Carvalho would represent a significant investment but one that could well safeguard the future of the club's midfield.
If Manchester United or Liverpool do intend on making a formal bid for Carvalho soon, however, then they will surely need to act fast as Monaco's vast wealth is such that they are unlikely to find it difficult to eventually meet his release clause.
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