The movement's spokespersons refused to say, however, whether they would give Carter a letter from Shalit.
Carter is expected to arrive in the Gaza Strip in the afternoon hours. He plans to meet with Hamas members and give them the letter he received from the kidnapped soldier's parents.
The Hamas organization appeared encouraged by Carter's visit, saying the meeting would serve as another layer in the world's openness towards the group.
Movement members said that the series of meetings held by the former US leader, who is considered close to President Barack Obama, and the fact that he European Union parliament members have been meeting in the past few months with the movement's men, show that the world and the American administration understand that there is no longer any use in the political siege on Hamas.
According to the sources, Hamas has become an element which cannot be ignored.
Hamas members added that the expected meeting with Carter would not be the last, and that public meetings with other senior American and EU officials were expected "very soon".
The international Quartet's Mideast envoy, former British Prime Minister Tony Blair, visited Gaza on Monday and called for an end to the blockade on the Strip.
Gilad Shalit was kidnapped into the Gaza Strip 1,087 days ago.