The United States has so far rebuffed repeated pleas from Saudi Arabia's Crown Prince Mohammed bin Salman to join the fight in Yemen.
The New York Times reported that the US administration had prepared to launch air strikes on Sunday against the Houthi rebels after fresh pleas from the Saudi crown prince.
But the newspaper said Trump called off the air strikes at the last minute, even as troops were loading bombs onto aircraft.
Reuters could not immediately verify the Times' report.
Rashid al-Alimi, the Riyadh-based president of Yemen's Saudi-backed internationally recognised government, asked Trump by phone on Sunday for US military help, four sources told Reuters.
Two of the sources said Trump made no direct pledge of support during the call.
Saudi Arabia was apparently struck on Saturday for the first time since the escalation began, with explosions heard and smoke visible near the airport.
The Houthi rebels said they had fired missiles at the Saudi capital while Saudi officials have not commented.
The Houthis, who have controlled Yemen's capital since 2014, seized the country's Red Sea coast this month in an advance that routed forces of the Saudi-backed government.
Saudi Arabia has launched hundreds of strikes in recent days on the fighters' positions in the highlands, the Houthis say.
According to the United Nations, nearly 700 people have been killed and thousands injured in the flare-up in fighting.
Nearly 130,000 Yemenis have fled homes within Yemen, and more than 3000 have fled by boat across the Red Sea to Africa.
Five Yemeni military sources told Reuters the Houthi militia were now seeking to seize strategic heights, known as the Kahboub Mountains, in the two provinces of Taiz and Lahij, which separate the Red Sea coast from the government-held south.
Fighting has centred on the outskirts of the remote mountain Al-Wazi'iyah district in Taiz province, and Ras al-Ara along the Indian Ocean coast between the Bab el-Mandeb Strait at the Red Sea's mouth and the government's base in Aden, the sources said.
"The government forces appear to remain largely in a defensive posture following the loss of the Bab el-Mandeb area while the Houthis are seeking to maintain the momentum of their advance despite being exposed to intensive air strikes," said Mohammed al-Qadhi, a veteran Yemeni journalist now working as a political analyst.
"These mountains could become critical to determining whether the Houthis succeed in consolidating their recent territorial gains or whether government forces will be able to retain the positions necessary to mount a counteroffensive."
The Houthis descended from their stronghold in northern Yemen, taking over the capital of Sanaa in September 2014 and forcing the country's government into exile in Saudi Arabia.
The following year, Saudi Arabia led a multi-national coalition that entered the war in an attempt to restore the government.
The coalition imposed an air and sea blockade on the Houthi-held areas as part of its war efforts.
with AP
