The Daily Star says Grant has been offered the chance to replace Steve McClaren following the former England coach's departure to German side Wolfsburg.
Portsmouth boss Grant has told pals he is torn between West Ham's offer of a £1.3m-a-year contract and the chance to manage in the Champions League again.
A source close to the former Chelsea boss said: "Avram was all set to join West Ham until he had the call from Holland.
"Joining West Ham would obviously be more lucrative, but Twente are Dutch champions and the opportunity to manage in Europe is appealing.
"Avram is weighing up the pros and cons of both offers and this is not a decision he will take lightly."