Palestinian Prime Minister Ismail Haniya has indicated that two foreign journalists kidnapped in Gaza City on 14 August may be freed soon.
A video was released of the pair on Wednesday
The two journalists, American Steve Centanni and New Zealander Olaf Wiig, both work for US organisation Fox News.
The group holding the men has demanded that the US free all Muslim prisoners in its jails in exchange for the pair.
A deadline set by the kidnappers for their demands to be met expired at midday (0900 GMT) on Saturday.
The kidnappers did not say what would happen if their demands were not met and the deadline passed without them making any public comment.
The group holding the journalists call themselves the Holy Jihad Brigades and have not been heard of before the abduction.
The BBC's Alan Johnston in Ramallah says that throughout the kidnapping the Palestinian authorities have seemed to be at a loss as to who was behind it.
The effort to free the foreigners seemed to be getting nowhere, our correspondent says, but senior figures in the Hamas-led government have now begun to sound much more positive.
Mr Haniya, has gone so far as to suggest that the kidnapping might even be resolved within hours, and although he revealed very little, he did say that promises had been given that the journalists would not be harmed.
These are by a long way the most promising comments from the government so far and the families of the kidnapped men will be desperately hoping that this affair may now be drawing to an end, our correspondent adds.
Over the past two years, a number of foreigners have been kidnapped in Gaza. All have been freed unharmed.