WASHINGTON - US President Barack Obama has “cleared the air” with Saudi King Salman during his visit to the kingdom despite reports of a cool reception in Riyadh, according to the White House .

Yet even as White House officials stressed that the leaders made progress, a prominent member of the Saudi royal family told CNN “a recalibration” of the US-Saudi relationship was needed amid regional upheaval, dropping oil prices and ongoing strains between the two longtime allies.

Statements after the meeting made clear that deep differences remain on several of these points, with the two sides agreeing to disagree and a US official characterizing the encounter as the start of a discussion rather than a venue for solutions. But the two leaders glossed over some of the thorniest matters, including a Saudi threat to dump US assets if Obama signs into law a bill that could make the kingdom liable for damages stemming from the September 11 terror attacks.

There is going to have to be “a recalibration of our relationship with America,” former Saudi Intelligence Chief Prince Turki Al-Faisal told CNN’s Christiane Amanpour. “How far we can go with our dependence on America, how much can we rely on steadfastness from American leadership, what is it that makes for our joint benefits to come together,” Turki said. “These are things that we have to recalibrate.”