Carter, who spent several days meeting with Israeli government officials, accepted the letter for Shalit from the kidnapped soldier's parents Friday, and promised to deliver it to Hamas, Maariv reported Tuesday.
Hamas leader Mahmoud Zahar told Israel Radio Tuesday, the organization will consider delivering the letter to the soldier.
Ynetnews.com said Hamas officials refused to say whether they would give Carter a letter from the soldier for his parents.
At a meeting with members of the Knesset's foreign affairs and defense committee Monday, Carter said he will do his utmost to receive "a sign of life" from Shalit, who has been held captive in Gaza for three years.
Knesset speaker Reuven Rivlin who met with Carter Monday, criticized Hamas for preventing anyone, including Red Cross officials, from visiting Shalit, the paper said.
Shimshon Liebman, a member of a committee canvassing for the soldier's return, told the newspaper the Shalit family hopes Carter will succeed in receiving some sign of the soldier's well being.
"We pray that former President Carter, one of a few Americans who maintain contact with Hamas, will succeed in returning with a sign of life from Gilad," he said.
In June 2006, while on operational duty near the Gaza border, Shalit was abducted and taken to Gaza.