The Argentina international, who is currently in Venezuela for his country's Copa America campaign, has been linked to several clubs in England and abroad but is keen to remain in the Premier League.
A statement from Tevez's representative Kia Joorabchian, said: "Inter put a deadline on the decision and Carlos' heart is really in the Premier League, so he was not prepared to meet that cut-off point."
The tenacious forward recovered from a difficult start to life in Britain following his move from Corinthians last summer to win over Hammers fans and neutrals alike, and was instrumental in the club's successful battle to avoid relegation.
West Ham are understandably keen to keep him and are involved in negotiations about his future.
Real Madrid, Liverpool and Inter were among the clubs strongly linked with a move for the 23-year-old but Inter have seemingly ruled themselves out of the running due to their demands.
Negotiations are ongoing with the Hammers which could see Tevez remain at Upton Park after all.
Joorabchian, who heads up Media Sports Investments, revealed reports that Tevez has been offered £100,000 by the Hammers to stay are "absolutely not true".
"He has not been offered that sum of money to stay, but it is true we are still talking and hope to come to some arrangement."