The US Wall Street Journal quoted a Saudi official as saying his country was considering proposals for direct talks with the Huthis.
The Saudi official told the newspaper that the kingdom did not want to drag into a long war in Yemen, but in return it did not want to appear "weak" or "damaged" in this issue, especially with the tension in the region because of the crisis with Iran.
The Wall Street Journal says the Houthis secretly offered to comment on their attacks on Saudi Arabia, a move diplomats saw as an indication of the willingness of some of the group's leaders to take a distance from Iran.
But Saudi Arabia remains skeptical about the true intentions of the Houthis and their ability to ensure a ceasefire because of internal disagreements over the real extent of the alliance with Tehran.
The newspaper said Riyadh was looking for a political solution to end the war in Yemen, which "tarnished" reputation in the United States because of the number of casualties in the war.
At the end of last month, a UN report - known as Al Jazeera, the Saudi-UAE alliance and the parties to the Yemeni conflict - kept a black list of violators of children's rights, documenting the killing and maiming of 1,689 children over the past year.
The report also confirmed that the Saudi-led coalition was responsible for killing 720 children, most of them because of air raids. Al-Jazeera correspondent said the United Nations pointed to the responsibility of the Houthis for the deaths of 398 victims.
The New York Times reported that the recent withdrawal of the UAE from several parts of Yemen made the war "a swamp for Saudi Crown Prince Mohammed bin Salman," suggesting that the Saudis might turn to Washington to compensate for the shortage. While diplomats saw the withdrawal as an incentive for them to negotiate with the Huthis.